5A 5B 5C 5D Arithmetic 5E 5F and geometric 5G 5H sequences 5Imathsbooks.net/JACPlus Books/12 Further Maths/Ch05... · 1. An arithmetic sequence is a sequence of numbers for which

202

AReAs oF sTudY

5A Recognition of arithmetic sequences 5B Finding the terms of an arithmetic

sequence 5C The sum of a given number of terms of

an arithmetic sequence 5D Recognition of geometric sequences 5E Finding the terms of a geometric

sequence 5F The sum of a given number of terms of

a geometric sequence 5G Applications of geometric sequences 5H Finding the sum of an infi nite geometric

sequence 5I Contrasting arithmetic and geometric

sequences through graphs

5

The recognition of arithmetic sequences and • the evaluation of terms and the sum of a fi nite number of terms and applicationsThe recognition of geometric sequences, the • evaluation of terms and the sum of a fi nite number of terms; applications, including growth models

Infi nite geometric sequences and the sum of an • infi nite geometric sequenceContrasting arithmetic and geometric sequences • through the use of graphs involving a discrete variable.

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Digital doc10 Quick Questions

Arithmetic and geometric sequences

introductionPatterns occur naturally in many real-life situations; for example the addition of interest to bank accounts, plant spacing in a winery and the stacking of logs in a pile. Two of the most common patterns are termed arithmetic and geometric sequences. Recognition of these two patterns is important in analysing situations that occur normally in the real world. Look at the sequence in the shaded column on this bank statement.

Date Description Debit Credit Balance

1.1.2006 Deposit 1000.00 1000.00

1.1.2007 Interest 100.00 1100.00

1.1.2008 Interest 100.00 1200.00

1.1.2009 Interest 100.00 1300.00

maths Quest 12 Further mathematics for the Casio Classpad

For this module (chapters 5 and 6), your CAS calculator will be used to solve problems or display a sequence graphically or in a list. The solve function on the Main screen, the NumSolve screen, the Sequence screen, the Spreadsheet screen and the Graph screen will be used.

203Chapter 5 Arithmetic and geometric sequences

Recognition of arithmetic sequencesA sequence in mathematics is an ordered set of numbers.

An arithmetic sequence is one in which:1. the difference between any two successive terms is the same, or2. the next term in the sequence is found by adding the same number.

Consider the arithmetic sequence:4, 7, 10, 13, 16, 19, 22.

The difference between each successive term is +3, or similarly, the next term is found by adding 3 to the previous term. We can see that a positive common difference gives a sequence that is increasing. We say that the common difference is +3, stated as d = +3.

4 73

103

133

163

193

223

The first term of the sequence is 4. We refer to the first term of a sequence as ‘a’. So in this example, a = 4.

In the arithmetic sequence above, the first term is 4, the second term is 7, the third term is 10 and so on. Another way of writing this is:

t1 = 4, t2 = 7 and t3 = 10.There are 7 terms in this sequence. Because there is a countable number of terms in the

sequence, it is referred to as a finite sequence.

5A

204

The arithmetic sequence below

37, 30,7

23,7

16,7

9...7

is an infinite sequence since it continues endlessly as indicated by the ellipsis (. . .) after the final term shown. The first term, a, is 37 and the common difference, d, is −7. We can see that a negative common difference gives a sequence that is decreasing.

1. An arithmetic sequence is a sequence of numbers for which the difference between successive terms is the same.

2. The first term of an arithmetic sequence is referred to as ‘a’.3. The common difference between successive terms is referred to as ‘d’.4. tn is the term number; for example, t6 refers to the 6th term in the sequence.

Which of the following are arithmetic sequences?a 7, 13, 19, 25, 31, . . . b −81, −94, −106, −120, −133, . . .c 1.3, 2.5, 3.7, 4.9, 6.3, . . . d −11

2, −1, − 1

2 , 0, 12 , . . .

Think WRiTe

a 1 Write the sequence. a 7, 13, 19, 25, 31, …

2 Calculate the difference between the first term, t1, and the second term, t2.

t2 − t1 = 13 − 7 = 6

3 Calculate the difference between the second term, t2, and the third term, t3.

t3 − t2 = 19 − 13 = 6

4 Calculate the difference between the third term, t3, and the fourth term, t4.

t4 − t3 = 25 − 19 = 6

5 Calculate the difference between t5 and t4.

t5 − t4 = 31 − 25 = 6

6 Check that the differences are the same and write your answer.

There is a common difference of 6, therefore d = 6.This is an arithmetic sequence.

b 1 Write the sequence. b −81, −94, −106, −120, −133, . . .

2 Calculate the difference between the first term, t1, and the second term, t2.

t2 − t1 = −94 − (−81) = −13

3 Calculate the difference between the second term, t2, and the third term, t3.

t3 − t2 = −106 − (−94) = −12

4 There is no need to do any further checks as the two differences are not the same.

There is no common difference.This is not an arithmetic sequence.

c 1 Write the sequence. c 1.3, 2.5, 3.7, 4.9, 6.3, . . .

2 Calculate the difference between t2 and t1. 2.5 − 1.3 = 1.2




6 Check that the differences are the same. There is no common difference.This is not an arithmetic sequence.

WoRked exAmple 1



d 1 Write the sequence. d −112, −1,

−12, 0, 1

2, …

2 Calculate the difference between t2 and t1.−1 − −11

2 = 12

3 Calculate the difference between t3 and t2.−1

2 − −1 = 12

4 Calculate the difference between t4 and t3. 0 − −1

2 = 1

2

5 Calculate the difference between t5 and t4.12 − 0 = 1

2

6 Check that the differences are the same. There is a common difference of 12.

This is an arithmetic sequence.

Write the value of a and d for each of the following arithmetic sequences.a 1.2, 3.6, 6, 8.4, 10.8, . . . b −12

5 , − 25, 3

5, 135 , 23

5, . . .

Think WRiTe

a 1 Write the sequence. a 1.2, 3.6, 6, 8.4, 10.8, …

2 What is the first term? a = 1.2

3 What is the difference between t2 and t1?You need check only once as the question states that this is an arithmetic sequence.

t2 − t1 = 3.6 − 1.2 = +2.4 d = +2.4

4 Write your answer. The arithmetic sequence has a first term, a, of 1.2 and a common difference, d, of +2.4.

b 1 Write the sequence. b −125,

− 25, 3

5 , 135 , 23

5 , . . .

2 What is the first term? a = −125

3 What is the difference between t2 and t1? t3 − t2 = − 2

5 − −125

= +1 d = +1

4 Write your answer. The arithmetic sequence has a first term, a, of −12

5 and a common difference, d, of +1.

WoRked exAmple 2

Given an arithmetic sequence, identify:1. the first term, a and the common difference, d.

Given an unspecified sequence, establish whether it is arithmetic by testing all terms 2. for a common difference: d = t2 − t1 = t3 − t2 = t4 − t3 = . . .

RememBeR

Recognition of arithmetic sequences 1 We 1a State which of the following are arithmetic sequences.

a 2, 7, 12, 17, 22, . . . b 2, 4, 8, 16, 32, . . . c 2, 4, 6, 8, 10, . . .d 3, 7, 11, 15, 20, . . . e 4, 8, 11, 15, 19, . . . f 3, 30, 300, 3000, 30 000, . . .g 1, 1, 2, 4, 8, . . . h 2, 2, 4, 4, 8, . . . i 10, 20, 30, 40, 50, . . .

2 We2 For those arithmetic sequences found in question 1, write the values of a and d.

exeRCise

5A

206

3 We 1b State which of the following are arithmetic sequences.

a −123, −23, 77, 177, 277, . . . b −1, 3, 1, 5, 3, . . .c −7, −1, 5, 11, 17, . . . d −67, −27, 13, 53, 93, . . .e 5, −2, −9, −16, −23, . . . f −7, −18, −29, −30, −39, . . .g 1, 0, −1, −3, −5, . . . h 0, −10, −21, −32, −43, . . .

4 For those arithmetic sequences found in question 3, write the values of a and d.

5 We 1c State which of the following are arithmetic sequences.

a 0.7, 1, 1.3, 1.6, 1.9, . . . b 2.3, 3.2, 5.2, 6.2, 7.2, . . .c −3.5, −2, −0.5, 1, 2.5, . . . d −2.7, −2.5, −1.7, −1.5, −0.7, . . .e 2, 0.1, −2.1, −3.3, −4.5, . . . f −5.2, −6, −6.8, −7.6, −8.4, . . .


7 We 1d State which of the following are arithmetic sequences.

a 12 , 11

2 , 212 , 31

2 , 412 , . . . b 1

4 , 34 , 11

4 , 1 34 , 21

4 , . . .

c 15 , 3

5 , 1, 125 , 14

5 , . . . d − 34 , 0, 3

4 , 112 , 21

4 , . . .

e 13 ,

− 13 , −1, −11

3 , 2, . . . f − 12 ,

− 14 ,

− 16 ,

− 18 ,

− 110 , . . .


9 Show which of the following situations are arithmetic sequences.a A teacher hands out 2 lollies to the

first student, 4 lollies to the second student, 6 lollies to the third student and 8 lollies to the fourth student.

b The sequence of numbers after rolling a die 8 times.

c The number of layers of paper after each folding in half of a large sheet of paper.

d The house numbers on the same side of a street on a newspaper delivery route.

e The cumulative total of the number of seats in the first ten rows in a regular cinema (for example, with 8 seats in each row, so there are 8 seats after the first row, 16 seats after the first 2 rows, and so on).

10 For those arithmetic sequences found in question 9, where appropriate information is given, write the value of a and d.

11 For the following arithmetic sequences:a 4, 13, 22, 31, . . . which term, tn, will be equal to 58?b 9, 4.5, 0, . . . which term, tn, will be equal to −18? c −60, −49, −38, . . . which term, tn, will be the first to be greater than 10? d 100, 87, 74, . . . which term, tn, will be the first to be less than 58?

12 Jenny receives 5 dollars for completing the first kilometre of a walkathon and 7 dollars more for completing each subsequent kilometre. Write the arithmetic sequence that represents the amount received by Jenny for each kilometre walked from 1 to 10 kilometres.

13 Each week, Johnny buys a pack of 9 basketball cards. In the first week Johnny has 212 cards in his collection. Give the total number of cards Johnny has for each of the first five weeks.

exAm Tip Students have to show two subtractions having or not having the same result. Only showing the calculations and not stating that the results are equal or not equal is not accepted. Some students merely show the results of the calculations without indicating where these results came from, or their significance.

[Assessment report 2 2007]



14 mC Which of the following could be the first five terms of an arithmetic sequence?A 1, 3, 9, 12, 15, . . . B −266, −176, −86, 4, 94, . . .C 3, 3, 6, 6, 9, . . . D 0, 1, 2, 4, 8, . . .E −3, −1, 0, 1, 3, . . .

15 mC 3, 10, 17, 24, 31, 38For the arithmetic sequence above, it is true to say that it is:A an infi nite sequence with a = 3 and d = 7 B an infi nite sequence with a = 7 and d = 3C an infi nite sequence with t2 = 10 and d = 7 D a fi nite sequence with a = 3 and d = 7E a fi nite sequence with a = 7 and d = 3.

Finding the terms of an arithmetic sequenceConsider the arithmetic sequence for which a = 8 and d = 10.

8 18 28 38 4810 10 10 10

Now, t1 = 8 t1 = at2 = 8 + 10 t2 = a + d t2 = a + 1dt3 = 8 + 10 + 10 t3 = a + d + d t3 = a + 2dt4 = 8 + 10 + 10 + 10 t4 = a + d + d + d t4 = a + 3dt5 = 8 + 10 + 10 + 10 + 10 t5 = a + d + d + d + d t5 = a + 4d

We notice a pattern emerging. That pattern can be described by the equation:tn = 8 + (n − 1) × 10

where n represents the number of the term.For example, if n = 4, then the fourth term is:

t4 = 8 + (4 − 1) × 10t4 = 8 + 3 × 10t4 = 38

Therefore, the 4th term is 38.We can generalise this rule for all arithmetic sequences.

tn = a + (n − 1) dwhere tn is the nth term a is the fi rst term d is the common difference.

This rule enables us to fi nd any term of an arithmetic sequence provided we know the value of a and d.

Find the 20th term of the following arithmetic sequence.5, 40, 75, 110, 145, . . .

Think WRiTe/displAY

Method 1: Using the rule

1 Find the value of a. a = 5

2 Find the value of d. You need to calculate only one difference as the question states that it is an arithmetic sequence.

d = t2 − t1 = 40 − 5 = 35

WoRked exAmple 3

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Interactivityint-0007

Number patterns

208

3 Use the rule tn = a + (n − 1) d where n is 20 for the 20th term.

t20 = 5 + (20 − 1) × 35t20 = 5 + 19 × 35t20 = 670

4 Write the answer. The 20th term is 670.

Method 2: Using a CAS calculator

This requires that you enter an equation to solve for one of the variables. The numeric solver will try to calculate the unknown value provided enough information has been entered.

1

2 Write your answer. The 20th term is 670.

If we are given only two terms of an arithmetic sequence, we are able to use the rule tn = a + (n − 1) d to set up two simultaneous equations to find the value of a and d and hence write the rule for the arithmetic sequence.

The third term of an arithmetic sequence is −1 and the fifth term is 11.a Write the rule for the arithmetic sequence. b Find the 50th term of the sequence.

Think WRiTe

a 1 We know that t3 = −1 and that tn = a + (n − 1) d.

a t3 = a + 2d = −1

2 We know that t5 = 11 and that tn = a + (n − 1) d.

t5 = a + 4d = 11

3 Solve the 2 equations simultaneously using the elimination technique.Eliminate a, by subtracting equation [1] from equation [2].

a + 2d = −1 [1] a + 4d = 11 [2] 2d = 12 [2] − [1] d = 6

4 Evaluate a by substituting d = 6 into either of the two equations.

Substituting d = 6 into [1]:a + 12 = −1 a = −13

WoRked exAmple 4


On the NumSolve screen, use the soft keyboard to complete the entry line as:t = a + (n − 1) × dThen press E.The list of variables will appear. Enter the known values and select the variable to solve by checking the adjacent bullet. Then tap:• 1 (on the tool bar)• OK


5 To fi nd the rule, substitute values for a and d into tn = a + (n − 1) d.

tn = −13 + (n − 1) × 6tn = −13 + 6n − 6tn = −19 + 6n

b 1 To fi nd the 50th term or t50, substitute n = 50 into the rule found.

b tn = −19 + 6nt50 = −19 + 6 × 50 = −19 + 300 = 281

2 Write your answer. The 50th term is 281.

If the fi rst three terms of an arithmetic sequence are 5.2, 7.4 and 9.6, which term is equal to 53.6?

Think WRiTe/displAY


1 Find the rule for the arithmetic sequence. a = 5.2 d = t2 − t1 = 7.4 − 5.2 = 2.2tn = 5.2 + (n − 1) × 2.2 = 5.2 + 2.2n − 2.2 = 3 + 2.2n

2 Which term is equal to 53.6?Substitute the values of a and d into tn = a + (n − 1) d.

53.6 = 3 + 2.2n

n = 50 62 2

..

n = 23

3 Write your answer. The term 53.6 is the 23rd term.


1

2 Write your answer. The term 53.6 is the 23rd term.

WoRked exAmple 5eBookpluseBookplus

Tutorialint-0446

Worked example 5

On the NumSolve screen, use the soft keyboard to complete the entry line as:t = a + (n − 1) × dThen press E.The list of variables will appear. Enter the known values and select the variable to solve by checking the adjacent bullet. Then tap:•1 (on the tool bar)• OK

210

An ant colony is studied and found to have a population of 10 000 in the first week of the study. The population increases by 500 each week after that.a Write a rule for the number of ants in the colony in week n of the study.b When will the ant population double in size?

Think WRiTe/displAY


a 1 We know that a = 10 000 and d = 500 and that tn = a + (n − 1) d.

a tn = 10 000 + (n − 1) × 500 = 10 000 + 500n − 500

2 Substitute and simplify. tn = 9500 + 500n

b 1 Using the rule found, we need to find which term is equal to 10 000 doubled or tn = 20 000.

b tn = 9500 + 500n20 000 = 9500 + 500n10 500 = 500n

n = 10 500

500 n = 21

2 Write your answer. The ant population will double to 20 000 in the 21st week.


1

WoRked exAmple 6


To define an arithmetic sequence on the Main screen, tap:• Action• Command• DefineComplete the entry line as:Define t(n) = 10 000 + (n − 1) × 500 Then press E.


2

3 Write your answer. The population will reach 20 000 in week 21.

To fi nd the terms of an arithmetic sequence, use the equation:tn = a + (n − 1)d

where tn is the nth term a is the fi rst term d is the common difference.

RememBeR

Finding the terms of an arithmetic sequence 1 We3 For each of the arithmetic sequences given, find:

a the 25th term of the sequence 2, 7, 12, 17, 22, . . .b the 19th term of the sequence 2, 4, 6, 8, 10, . . .c the 30th term of the sequence 0, 100, 200, 300, 400, . . .d the 27th term of the sequence −7, −1, 5, 11, 17, . . .e the 33rd term of the sequence 5, −2, −9, −16, −23, . . .f the 39th term of the sequence 14.1, 28.2, 42.3, 56.4, 70.5, . . .

2 We4 Evaluate the following.a The 2nd term of an arithmetic sequence is 13 and the 5th term

is 31. What is the 17th term of this sequence?b The 2nd term of an arithmetic sequence is −23 and the 5th term

is 277. What is the 20th term of this sequence?c The 2nd term of an arithmetic sequence is 0 and the 6th term is −8. What is the 32nd term

of this sequence?d The 3rd term of an arithmetic sequence is 5 and the 7th term is −19. What is the 40th term

of this sequence?

3 We5 Evaluate the following.a The fi rst 3 terms of an arithmetic sequence are 3, 9 and 15. Which term is equal to 141?b The fi rst 3 terms of an arithmetic sequence are −9, −6 and −3. Which term is equal to 72?c The fi rst 3 terms of an arithmetic sequence are 1.7, 2.5 and 3.3. Which term is equal to

28.1?d The fi rst 3 terms of an arithmetic sequence are −1.5, −2 and −2.5. Which term is equal to

−140.5?

exeRCise

5B

eBookpluseBookplus

Digital docsSkillSHEET 5.1

Solving simultaneous equations using the elimination method

SkillSHEET 5.2Solving linear

equations (arithmetic terms)

To solve for n, tap:• Interactive• Advanced• solveComplete the equation box as: t(n) = 20 000 Enter n in the variable box and then tap OK.

212

4 We6 A batsman made 23 runs in his first innings, 33 in his second and 43 in his third. If he continued to add 10 runs each innings, write a rule for the number of runs he would have made in his nth innings.

5 In a vineyard, rows of wire fences are built to support the vines. The length of the fence in row 1 is 40 m, the length of the fence in row 2 is 43 m, and the length of the fence in row 3 is 46 m. If the lengths of the fences continue in this pattern, write down a rule for the length of a fence in row number n.

6 A marker is placed 15 m from a white line by a P.E. teacher. The next marker is placed 25 m from the white line and the next 35 m from the white line. The teacher continues placing markers in this pattern.a Write a rule for the distance of marker n from the white line.b How many markers will need to be placed before the last marker is at least 100 metres

from the line?

7 mC The 41st term of the arithmetic sequence −4.3, −2.1, 0.1, 2.3, 4.5, . . . is:A 83.7 B 85.9 C 92.3 D 172.4 E 178.5

8 mC The 2nd term of an arithmetic sequence is −2 and the 5th term is 2.5. The 27th term of this sequence is:A 32.5 B 35.5 C 42.5 D 89.5 E 96

9 mC The numbers 8, 1 and −6 form the first three terms of an arithmetic sequence. In this arithmetic sequence the term which is equal to −258 is the:A 30th B 32nd C 37th D 39th E 42nd

10 Find the 28th term of the arithmetic sequence −5.2, −6, −6.8, −7.6, −8.4, . . . 11 Find the 31st term of the arithmetic sequence 1

5, 35, 1, 12

5, 145, . . .

12 The 3rd term of an arithmetic sequence is −16 and the 5th term is −4.2. What is the 19th term of this sequence?

13 The 4th term of an arithmetic sequence is 312 and the 7th term is 61

2. What is the 25th term of this sequence?

14 The 3rd term of an arithmetic sequence is 15 and the 8th term is 45. Which term of the sequence is equal to 183?

15 The 2nd term of an arithmetic sequence is 1 and the 6th term is −15. Which term of the sequence is equal to −167?

16 mC The 3rd term of a sequence is −1 and the fifth is 14. The term which is equal to 141.5 is the:A 9th B 11th C 18th D 20th E 22nd

17 Peter plants his first tomato seedling 0.5 m from the fence, the next 1.3 m from the fence and the next 2.1 m from the fence. If he continues to plant in this pattern, how far will the 14th seedling be from the fence?



18 Olivia began her china collection in 1951. She was given 3 pieces of china that year and added 2 pieces each year after that. How many pieces did Olivia have in her collection in the year 2000?

19 The membership of a local photography club was 7 in its first year. If the club added 4 members to its membership each year, write a rule for the number of members in the club in year n.

20 The first fence post in a fence is 12 m from the road, the next is 15.5 m from the road and the next is 19 m from the road. The remainder of the fence posts are spaced in this pattern.a Write a rule for the distance of fence post n from the road.b If 100 posts are to be erected, how far will the last post be from the road?

The sum of a given number of terms of an arithmetic sequenceWhen the terms of an arithmetic sequence are added together, an arithmetic series is formed.So, 5, 9, 13, 17, 21, . . . is an arithmetic sequencewhereas 5 + 9 + 13 + 17 + 21 + . . . is an arithmetic series.The sum of n terms of an arithmetic sequence is given by Sn.

Consider the finite arithmetic sequence below.1, 2, 3, 4, 5, 6, 7, 8, 9, 10

The sum of this arithmetic sequence is given by S10 since there are 10 terms in the sequence.So, S10 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

Note that the sum of the first and last terms is 11. Also, the sum of the second and second last terms is 11. Similarly, the sum of the third and third last term is 11. This pattern continues with the fourth and fourth last terms as well as with the fifth and fifth last terms. There are in fact five lots of 11.

We can formalise this pattern to obtain a rule which applies to all arithmetic sequences.Let Sn = a + (a + d ) + (a + 2d ) + . . . + (l − 2d ) + (l − d ) + lwhere l is the last term of the sequence.

By reversing the order of the series above, we obtainSn = l + (l − d ) + (l − 2d ) + . . . + (a + 2d ) + (a + d ) + a

By adding these two equations, we obtain2Sn = (a + l) + (a + d + l − d ) + (a + 2d + l − 2d ) + . . . (l − d) + (a + d) + (a + l)2Sn = (a + l) + (a + l) + (a + l) + . . . (a + l) + (a + l)2Sn = n(a + l) where n represents the number of terms in the sequence.

So, Sn = +1

2n a l( )

The sum of n terms of an arithmetic sequence with a as its first term and l as its last term is given by

Sn

a ln = ( + )2

5C

214

Recall that the nth term of an arithmetic sequence is given by

tn = a + (n − 1)d

So, for the sum of n terms, l is the last term; that is, tn = l.So, the last term is

l = a + (n − 1)d

Substituting this into Sn = +na l

2( )

we obtain Sn = + + −na a n d

21{ [ ( ) ]}

= + −n

a n d2

2 1[ ( ) ]

An alternative formula for the sum of n terms of an arithmetic sequence when the value of a and d are known, is given by

Sn

a n dn = + −2

[2 ( 1) ]

Find the sum of the first ten given terms of the arithmetic sequence4, 10, 16, 22, 28, 34, 40, 46, 52, 58.

Think WRiTe/displAY


1 We know the values of the first and last term and that there are ten terms in the series.

a = 4 l = 58n = 10

2 Use the series formula Sn = n a l2

( )+ . Sn = n

a l2

( )+

S10 = 102

(4 58)+

= 5 × 62 = 310


1 We know the value of a and d and n. a = 4d = 10 − 4 = 6n = 10

2 Use the formula Sn = n

a n d2

[2 ]+ ( −1) . Sn = n

a n d2

[2 ]+ ( −1)

Sn = 102

[2 ]× + −4 10 1 6( )

S10 = 5[8 + 9 × 6] = 5[8 + 54] = 5 × 62 = 310

3 Write the answer. The sum of the first 10 terms is 310.

WoRked exAmple 7




1

2

3 Write your answer. The sum of the fi rst 10 terms is 310.

The fi rst term of an arithmetic sequence is 5 and the seventh term is 29. What is the sum of the fi rst 10 terms of this sequence?

Think WRiTe

1 Find the value of a and d.The value of a is given.To fi nd the common difference, d, use the formula tn = a + (n − 1)d, as we know the value of the 7th term is 29.

a = 5 tn = a + (n − 1)d t7 = 5 + 6 × d = 295 + 6d = 29 6d = 24 d = 4

2 Use the formula Sn = n2

[2a + (n − 1)d ]. S10 = 102

[2 × 5 + (10 – 1)4]

S10 = 5[10 + 9 × 4] = 230

3 Write your answer. If a = 5 and d = 4, the sum of the fi rst ten terms is 230.


Tutorialint-0447

Worked example 8

To defi ne an arithmetic sum on the Main screen, tap:• Action• Command• Defi neComplete the entry line as:

Defi ne s(n) = n2

(2a + (n − 1)d )

Then press E.

To calculate the sum of the fi rst 10 terms, complete the entry line as:s(n) | a = 4 | d = 6 | n = 10Then press E.

216

The 3rd term of an arithmetic sequence is 4 and the 8th is −11. What is the sum of the fi rst 30 terms of the sequence?

Think WRiTe

1 Find out the value of a and d by setting up two simultaneous equations using tn = a + (n − 1)d.

t3 = a + 2d = 4 [1] t8 = a + 7d = −11 [2]

2 Eliminate a by subtracting equation 1 from equation 2.

a + 2d = 4 [1] a + 7d = −11 [2] 5d = −15 [2] – [1] d = −3 Substitute d = −3 into equation [1]. a + 2d = 4

a + 2 × −3 = 4 a – 6 = 4 a = 10

3 Use the formula Sn = n2

[2a + (n − 1)d ]. S30 = 302

[2 × 10 + (30 – 1)(−3)]

S30 = 15[20 + 29 × −3]S30 = −1005

4 Write your answer. If a = 10 and d = −3, the sum of the fi rst 30 terms is −1005.

WoRked exAmple 9

The fi rst term of a sequence is −7 and the sum of the fi rst 25 terms is 1625. Find:a the 25th term b the fi rst fi ve terms of the sequence.

Think WRiTe/displAY

a 1 We know a = −7 and S25 = 1625 and the 25th term is the last term, l.

Use the formula Sn = n2

(a + l ).

a S25 = 252

(−7 + l )

= 162512.5(−7 + l ) = 1625

−7 + l = 162512.5

−7 + l = 130 l = 137

2 Write your answer. l = t25 = 137The 25th term is 137.

b 1 To fi nd the common difference, d,

we can use Sn = n2

[2a + (n − 1)d ]

or tn = a + (n − 1)d having found

that the 25th term is 137.

b S25 = 252

[−14 + (25 − 1)d ]

= 162512.5 [−14 + 24d ] = 1625 −14 + 24d = 130 24d = 144 d = +6

WoRked exAmple 10



or tn = a + (n − 1)d t25 = 137

137 = −7 + (25 − 1)d 144 = 24d d = +6

2 The sequence is −7, −1, 5, 11, 17, . . .

A 1. series is the sum of terms in a sequence.S2. n is the sum of the fi rst n terms in a series; for example, S25 represents the sum of the fi rst 25 terms.Given the number of terms in an arithmetic series, 3. n, the fi rst term, a, and the last term, l, use Sn =

n2

(a + l).

Given the number of terms in an arithmetic series, 4. n, the fi rst term, a, and the common difference, d, use Sn =

n2

[2a + (n − 1)d ].

RememBeR

The sum of a given number of terms of an arithmetic sequence 1 We 7 For each of the given series, find:

a the sum of the fi rst 20 terms of the sequence 3, 7, 11, 15, 19, . . .b the sum of the fi rst 34 terms of the sequence 10, 20, 30, 40, 50, . . .c the sum of the fi rst 23 terms of the sequence −4, −1, 2, 5, 8, . . .d the sum of the fi rst 29 terms of the sequence 10, 7, 4, 1, −2, . . .

2 We8 The first term of an arithmetic sequence is 5 and the second is 9. Find the sum of

the first 40 terms of the sequence.

3 The first term of an arithmetic sequence is 0.7 and the second is 1. Find the sum of the first 25 terms of the sequence.

4 We9 For each of the following, evaluate the sum of a series, Sn.a The 3rd term of an arithmetic sequence is 19 and the 4th is 25. Find the sum of the fi rst

15 terms of the sequence.b The 2nd term of an arithmetic sequence is 3.6 and the 5th is 10.8. Find the sum of the fi rst

23 terms of the sequence.

exeRCise

5C

We know that a = −7 and d = +6.

Note: Alternately, on a CAS calculator, enter the fi rst term and press E. Add the common difference and then press E for each successive term.

218

c The 3rd term of an arithmetic sequence is −0.5 and the 6th is 4. Find the sum of the fi rst 26 terms of the sequence.

d The 2nd term of an arithmetic sequence is 0.75 and the 5th is 2.25. Find the sum of the fi rst 27 terms of the sequence.

5 We 10 The first term of an arithmetic sequence is 2 and the sum of the first 19 terms of the sequence is 551. Find:a the 19th termb the fi rst 3 terms of the sequence.

6 The first term of an arithmetic sequence is −4 and the sum of the first 30 terms of the sequence is 2490. Find:a the 30th termb the fi rst 3 terms of the sequence.

7 mC The sum of the first 21 terms of the sequence, 0, −312, −7, −10

12, −14, . . . is:

A −1470 B −735 C −700 D 36.75 E 735 8 mC The first term of an arithmetic sequence is −5.2 and the second is −6. The sum of the first

22 terms of the sequence is:A −598.4 B −299.2 C −242 D −70.4 E 70.4

9 What is the sum of the first 19 terms of the sequence −180, −80, 20, 120, 220, . . . ? 10 What is the sum of the first 28 terms of the sequence

− 1

2, 1

2, 11

2, 21

2, . . . ?

11 The 2nd term of an arithmetic sequence is 28.2 and the 6th is 84.6. Find the sum of the first 40 terms of the sequence.

12 The 1st term of an arithmetic sequence is 5.5 and the sum of the first 18 terms of this sequence is 328.5. Find:a the 18th termb the fi rst 3 terms of the sequence.

13 The 1st term of an arithmetic sequence is 11 and the sum of the first 20 terms of this sequence is −350. Find:a t20

b the fi rst 3 terms of the sequence.

14 Sam makes $100 profit in his first week of business. If his profit increases by $75 each week, what would his total profit be by the end of week 15?

15 George’s salary is to start at $36 000 a year and increase by $1200 each year after that. How much will George have earned in total after 10 years?

16 A staircase is designed so that the height of each step increases by 0.8 cm for each step. If the height of the first step is 15 cm, what is the total height of the first 17 steps?

17 Paula collects stamps. She bought 250 in the first month to start her collection and added 15 stamps to the collection each month thereafter. How many stamps will she have collected after 5 years?

18 Proceeds from the church fete were $3000 in 1981. In 1982 the proceeds were $3400 and in 1983 they were $3800. If they continued in this pattern:a what would be the proceeds from the year 2000 fete?b how much in total would the proceeds from church fetes from 1981 to 2000 have

amounted to?

19 Fees for groups meeting at the community centre rise by $5 each year. If the fees started at $60 a year in the first year:a how much will the fees be in the 20th year?b how much would a group which had met at the centre for all of

those 20 years have paid in total?

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Recognition of geometric sequencesA sequence in mathematics is an ordered set of numbers. A geometric sequence is one in which the fi rst term is multiplied by a number, known as the common ratio, to create the second term which is multiplied by the common ratio to create the third term, and so on. The fi rst term in a geometric sequence is referred to as a and the common ratio is referred to as r.

Consider the geometric sequence where a = 1 and r = 3. The terms in the sequence are

1 33

9 27 81...3 3 3

To discover the common ratio, r, of a geometric sequence you need to calculate the ratio of

successive terms, namely tt2

1. You could alternatively calculate

tt3

2 or

tt4

3 and so on.

A geometric sequence is a sequence of numbers for which the ratio of successive terms is the same.

tt

tt

tt

2

1

3

2

4

3= = = . . . = common ratio

The fi rst term of a geometric sequence is referred to as a.The common ratio between a term and its preceding term is referred to as r.

Which of the following are geometric sequences? For those that are geometric, state the values of a and r.a 2, 10, 50, 250, 1250, . . . b 4, −8, 16, −32, 64, . . .c −2, −6, 18, 54, −162, . . . d 6, 3, 3

2, 3

4, 3

8, . . .

Think WRiTe

a 1 Write the sequence. a 2, 10, 50, 250, 1250, . . .

2 Calculate the ratio of tt2

1.

tt2

1

102

=

= 5


2.

tt3

2

5010

=

= 5


3.

tt4

3

25050

=

= 5


4.

tt5

4

1250250

=

= 56 Check that all ratios are the same. There is a common ratio of 5.

This is a geometric sequence where a = 2 and r = 5.

b 1 Write the sequence. b 4, −8, 16, −32, 64, . . .


1.

tt2

1

84

=−

= −2

WoRked exAmple 11

5d

220


2.

tt3

2

168

= −

= −2


3. t

t4

3

3216

=−

= −2


4.

tt5

4

6432

= −

= −26 Check that all ratios are the same. There is a common ratio of −2.

This is a geometric sequence where a = 4 and r = −2.

c 1 Write the sequence. c −2, −6, 18, 54, −162, . . .


1. t

t2

1

62

=−

−

= +3


2.

tt3

2

186

= −

= −34 There is no need to do any further check

as the two ratios are not the same.There is no common ratio.This is not a geometric sequence.

d 1 Write the sequence. d 6, 3, 32

, 34

, 38

, . . .

2 Calculate tt2

1.

tt2

1

36

=

= 12

3 Calculate tt3

2.

tt3

2

32

3= ÷

= ×3

213

= 12

4 Calculate tt4

3.

tt4

3

34

32

= ÷

= ×34

23

= 24

= 12

5 Calculate tt5

4.

tt5

4

38

34

= ÷

= ×38

43

= 12

6 Check that all ratios are the same. There is a common ratio of 12

.

This is a geometric sequence where a = 6 and r = 12

.



Given a geometric sequence, identify:1. the fi rst term, a, and the common ratio, r =

tt2

1.

Given an unspecifi ed sequence, establish whether it is geometric by testing all terms 2.

for a common ratio, rtt

tt

tt

= = == = = =2

1

3= = =3= = =2

4

3. . .

RememBeR

Recognition of geometric sequences 1 We 11 State which of the following are geometric sequences.

a 1, 2, 4, 8, 16, . . . b 2, 6, 18, 54, 162, . . .c 1, 4, 16, 64, 256, . . . d 2, 6, 12, 24, 48, . . .e 1, 5, 25, 100, 125, . . . f 3, 6, 12, 24, 48, . . .g 5, 10, 20, 40, 80, . . . h 2, 3, 4, 5, 6, . . .

2 Look again at the geometric sequences found in question 1. Write the values of a and r.

3 State which of the following sequences are geometric sequences.a −1, −2, −4, −8, −16, . . . b −2, −4, 8, 16, −32, . . .c 0, −3, −9, −27, −81, . . . d −2, −6, −18, −54, −162, . . .e −4, −8, −16, −32, −64, . . . f −9, −3, 1, 3, 9, . . .g 1, 5, −25, 125, −625, . . . h −5, −20, −80, −320, −1280, . . .

4 For those geometric sequences found in question 3, write the value of a and r.

5 State which of the following sequences are geometric sequences.a 1, −2, 4, −8, 16, . . . b 1, 3, −9, 27, −81, . . .c 4, −12, 36, −108, 324, . . . d 7, −7, 7, −7, 7, . . .e 3, −6, 18, −54, 112, . . . f −2, 8, −12, 24, −48, . . .


7 State which of the following sequences are geometric sequences.a −2, 4, −8, 16, −32, . . . b −1, −5, 10, −20, −40, . . .c −3, −15, −75, −375, −1875, . . . d −10, −1, 0, 1, 10, . . .e 0, −4, −16, −64, −256, . . . f −6, 60, −600, 6000, −60 000, . . .


9 State which of the following sequences are geometric sequences.a 1.2, 2.4, 4.8, 9.6, 19.2, . . . b 2.3, 6.9, 13.8, 27.6, 55.2, . . .c 2.25, 4.5, 9, 18, 36, . . . d 7, 3.5, 1.75, 0.875, 0.4375, . . .e 10, 12, 14.4, 17.28, 20.736, . . . f −1.2, 3.6, 10.8, 21.6, 43.2, . . .


11 State which of the following sequences are geometric sequences.

a 12, 1

4, 1

8, 1

16, 1

32, . . . b 1

2, 1

6, 1

12, 1

18, 1

24, . . .

c 14, 1

20, 1

100, 1

500, 1

2500, . . . d 1

3, 2

3, 1

13, 2

23, 5

13, . . .

e 15, 1

10, 1

15, 1

20, 1

25, . . . f 1,

− 13

, 19,

− 127

, 181

, . . .


exeRCise

5d

222

13 mC Which of the following is a geometric sequence?A 1, 3, 6, 9, 12, . . . B 0.002, 0.02, 0.2, 20, 200, . . .C 9, −9, 3, −3, 1, . . . D 5, −5, 5, −5, 5, . . .

E 18, 1

7, 1

6, 1

6, 1

5, 1

4, . . .

14 mC There is a geometric sequence for which a is positive and r = −2. It is true to say that:A only one term of the sequence is a positive numberB the third term will be a negative numberC the third term will be less than the second termD the 5th term would be greater than the 6th termE the 4th term would be greater than the third term

15 mC There is a geometric sequence for which every term is negative. It could be said with certainty that:A a and r are both positive B a and r are both negativeC a is positive and r is negative D a is negative and r is positiveE a is greater than r

16 mC There is a geometric sequence for which every odd-numbered term is positive and every even-numbered term is negative. It could be said with certainty that:A a and r are both positive B a and r are both negativeC a is positive and r is negative D a is negative and r is positiveE a is greater than r

17 A savings account balance at the end of each of the past four years is given as follows: $100.00, $110.00, $121.00, $133.10.a Prove this is a geometric sequence.b State the value of a and r.

18 On the first day Jenny hears a rumour. On the second day, she tells two friends. On the third day, each of these two friends tell two of their own friends, and so on.a Write the geometric sequence for the fi rst fi ve days of this real-life situation.b Find the value of r.c How many people are told of the rumour on the 12th day?



19 Decay of radioactive material is modelled as a geometric sequence where r = 1

2. If there are 20 million radioactive

atoms, write the first 7 terms of the sequence. 20 Copy and complete the following.

a In a sequence that grows and where each term is positive, the r value is than 1.

b In a sequence that decays and where each term is positive, the r value is between

and .c In a sequence like 2, 2, 2, 2, 2, . . . , the r value

is 1.d In a sequence like 2, −6, 18, −54, . . ., the r value

is less than .e In a sequence like −54, 18, −6, 2, −1, . . . ,

the r value is between and .

Finding the terms of a geometric sequenceConsider the finite geometric sequence of seven terms for which a = 3 and r = 4.

3 124

48 192 3844 4 4

Now, t1 = 3 t1 = a t2 = 3 × 4 t2 = a × r t2 = a × r1

t3 = 3 × 4 × 4 t3 = a × r × r t3 = a × r 2

t4 = 3 × 4 × 4 × 4 t4 = a × r × r × r t4 = a × r 3

t5 = 3 × 4 × 4 × 4 × 4 t5 = a × r × r × r × r t5 = a × r 4 and so on . . .We notice a pattern emerging. That pattern can be described by the equation:

tn = 3 × 4n − 1.For example, if n = 5, t5 = 3 × 44.We can generalise this rule for all geometric sequences.

tn = ar n − 1

where tn is the nth term a is the first term r is the common ratio.This rule enables us to find any term of a geometric sequence provided we know the value of a and r.

Find the 12th term of the geometric sequence:2, 10, 50, 250, 1250, . . .

Think WRiTe/displAY


1 Find the value of a. a = 2

2 It has been stated that it is a geometric sequence, so find the value of r.

r = 10

2

= 5

WoRked exAmple 12

5e

224

The 2nd term of a geometric sequence is 8 and the 5th is 512. Find the 10th term of this sequence.

Think WRiTe

1 We know that t2 = 8 and that tn = a × r n − 1. t2 = a × r1

= 8

2 We know that t5 = 512 and that tn = a × r n − 1.

t5 = a × r4

= 512

3 Solve the 2 equations simultaneously by eliminating a, to fi nd r.Divide equation 2 by equation 1.

a × r1 = 8 [1] a × r4 = 512 [2]a ra r×a r×a r×a r×a r

=4 512

8 [2] ÷ [1]

r3 = 64 r = 4

4 To fi nd a, substitute the value of r. Substitute r = 4 into equation [1].a × 4 = 8 a = 2

5 Write the rule. tn = 2 × 4n − 1

6 To fi nd the 10th term, let n = 10. t10 = 2 × 49

= 524 288

7 Write your answer. The 10th term in the sequence is 524 288.

WoRked exAmple 13

3 Use the rule tn = a × r n − 1 to fi nd the 12th term.

t12 = 2 × 512 − 1

= 97 656 250

4 Write the answer. The value of the 12th term is 97 656 250.


1

2 Write your answer. The 12th term is 97 656 250.


To defi ne a geometric sequence on the Main screen, tap:• Action• Command• Defi neComplete the entry line as:Defi ne t(n) = a × r^(n − 1)Then press E.To calculate the 12th term, complete the entry line as:t(12) | a = 2 | r = 5Then press E.


The fi rst three terms of a geometric sequence are 2, 6, and 18.Which numbered term would be the fi rst to exceed 1 000 000 in this sequence?

Think WRiTe/displAY

Method 1: Using logarithms1 To fi nd the rule for the sequence, fi nd

a and r and substitute them into tn = a × r n − 1.

a = 2 r =

tt2

1

= 62

= 3 tn = 2 × 3n − 1

2 Set up the equation to be solved. 2 × 3n − 1 = 1 000 000 3n − 1 = 500 000

3 Express 3n − 1 and 500 000 in terms of logarithms to the base 10.

log10 (3n − 1) = log10 (500 000)(n − 1) log10 (3) = log10 (500 000)

(n − 1) = log ( )

log ( )10

10

500 000

3 n − 1 = 11.9445

n = 12.9445

4 The next whole number term is the 13th. The 13th term would be the fi rst to exceed 1 000 000.

Method 2: Using a CAS calculator1

2

3 The next whole number is 13. The 13th term would be the fi rst to exceed 1 000 000.


Tutorialint-0448

Worked example 14

To defi ne a geometric sequence on the Main screen, tap:• Action• Command• Defi neComplete the entry line as: Defi ne t(n) = a × r^(n − 1) | a = 2 | r = 3Then press E.

To solve for n when t(n) = 1 000 000, tap:• Interactive• Advanced• solveComplete the equation box as:t(n) = 1 000 000 Enter n in the variable box. Check ‘Solve numerically’ and tap OK.

226

WoRked exAmple 15

A real estate agent records the number of blocks of land to be sold on a new estate each week.If the number of blocks of land sold continue to follow a geometric sequence:a prove it is a geometric sequence and thus write

the value of the common ratio, rb write a rule for the number of blocks of land, tn,

sold in week number nc calculate the number of blocks of land sold in

week number 8.

Week number Number of blocks of land sold

1 128

2 64

3 32

Think WRiTe

a The first term, a, is 128. To find r, evaluate the ratios.

att2

1

64128

=

tt3

2

3264

=

= 12

= 12

The ratios are the same, so the terms follow a geometric sequence. The ratio,

r = 12

.

b Use tn = ar n − 1 to write the rule. b tn = 128 × 12

1

n−

c 1 Use the formula in part b . c t8 = 128 × 12

7

= 1

2 Write your answer. The number of blocks of land sold in the 8th week will be 1.

To find the terms of a geometric sequence, use the equation:tn = ar n − 1

where tn is the nth term a is the first term r is the common ratio.

RememBeR

Finding the terms of a geometric sequence 1 We 12 Find the value of the term specified for the given geometric sequences.

a Find the 10th term of the geometric sequence 2, 12, 72, 432, 2592, . . .b Find the 11th term of the geometric sequence 5, 35, 245, 1715, 12 005, . . .c Find the 18th term of the geometric sequence 8, 16, 32, 64, 128, . . .

exeRCise

5e



d Find the 8th term of the geometric sequence 11, 22, 44, 88, 176, . . .e Find the 11th term of the geometric sequence 5, 15, 45, 135, 405, . . .f Find the 15th term of the geometric sequence 2, 8, 32, 128, 512, . . .

2 Find the value of the term specified for the given geometric sequences in decimal form.a Find the 20th term of the geometric sequence 1.1, 2.2, 4.4, 8.8, 17.6, . . .b Find the 10th term of the geometric sequence 2.3, 2.76, 3.312, 3.9744, . . .c Find the 8th term of the geometric sequence 3.1, 8.06, 20.956, 54.4856, 141.662 56, . . .

3 Find the value of the term specified for the given geometric sequences in negative form.a Find the 9th term of the geometric sequence −2, −8, −32, −128, −512, . . .b Find the 12th term of the geometric sequence −6, −18, −54, −162, −486, . . .

4 We 13 Find the value of the term specified for the specified geometric sequences.a The 2nd term of a geometric sequence is 6 and the 5th term is 162.

Find the 10th term.b The 2nd term of a geometric sequence is 6 and the 5th term is 48.

Find the 12th term.c The 4th term of a geometric sequence is −32 and the 7th term

is −256. Find the 14th term.

5 We 14 Evaluate the following.a The fi rst three terms of a geometric sequence are 5, 12.5 and 31.25.

Which term would be the fi rst to exceed 50 000?b The fi rst three terms of a geometric sequence are 3.2, 9.6 and 28.8.

Which term would be the fi rst to exceed 1 000 000?c The fi rst three terms of a geometric sequence are 5.1, 20.4 and 81.6.

Which term would be the fi rst to exceed 100 000?

6 We 15 The number of cells of a micro-organism, after each process of cell division, can be summarised as follows.

1, 2, 4, 8, 16If the number of cells after each division continue to follow a geometric sequence, fi nd:a a rule for the number of cells after

n divisionsb the number of cells after 12 divisions.

7 A small town is renowned for spreading rumours. All of its citizens are aware in a short time of any new rumours. The spread of the rumour can be summarised in the table below right. If the number of citizens who have been told the rumour each day continues to follow a geometric sequence, fi nd:a a rule for the number of citizens on day nb the number of citizens told of the rumour

by day 5c on which day all 4230 citizens will know

of the rumour.

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Solving non-linear simultaneous

equations

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Solving indicial equations

exAm Tip A difference equation is not appropriate here as a formula in terms of n is required.


Day numberNumber of citizens in

the know1 12 63 36

228

8 Find the 9th term of the geometric sequence −6, 9, −13.5, 20.25, −30.375, . . . 9 Find the 8th term of the geometric sequence −6.2, 9.3, −13.95, 20.925, −31.3875, . . . 10 Find the 10th term of the geometric sequence 1

4, 1

8, 1

16, 1

32, 1

64, . . .

11 Find the 11th term of the geometric sequence 13, 2

3, 11

3, 22

3, 51

3, . . .

12 The 4th term of a geometric sequence is −81 and the 7th term is 2187. Find the 12th term.

13 The 4th term of a geometric sequence is 0.875 and the 7th term is 0.109 375. Find the 10th term.

14 The 3rd term of a geometric sequence is 1 and the 6th term is 827

. Find the 9th term.

15 The takings at a new cinema each month are recorded. If the takings each month continue to follow a geometric sequence, find:a a rule for the takings in month nb the takings in month 9.

16 A tomato grower has recorded the average height of his tomato bushes. If the height of the tomato bushes each year continues to follow a geometric sequence, find:a a rule for the height of the bushes in

year nb the height of the tomato bushes in year 5 (correct to 2 decimal places)c the year in which the height of the tomato bushes will exceed 2 m.

17 mC The 12th term of the geometric sequence 21, 63, 189, 567, . . . is:

A 6804 B 413 343 C 1 240 029 D 3 720 087 E 5 931 980 229

18 mC The 10th term of the geometric sequence 5, −10, 20, −40, 80, . . . is:

A −2560 B −1280 C 1280 D 5120 E 3 906 250

19 mC The 3rd term of a geometric sequence is −75 and the 6th term is −9375. The 9th term is:

A −5 859 375 B −1 171 875 C −32 805 D 32 805 E 234 375

20 mC The first three terms of a geometric sequence are 5.5, 7.7 and 10.78. The first term to exceed 100 would be the:

A 8th B 9th C 10th D 11th E 12th

The sum of a given number of terms of a geometric sequenceWhen the terms of a geometric sequence are added, a geometric series is formed. So 3, 6, 12, 24, 48, . . . is a geometric sequence, whereas 3 + 6 + 12 + 24 + 48 + . . . is a geometric series.

The sum of n terms of a geometric sequence is given by Sn.Consider the general geometric sequence a, ar, ar 2, ar 3, . . . ar n − 1.

Now, Sn = a + ar + ar 2 + ar 3 + . . . + ar n − 1

Also, multiplying each term by r, rSn = ar + ar 2 + ar 3 + ar 4 + . . . + ar n

So, rSn − Sn = −a + ar n since all the other terms cancel out.So, Sn (r − 1) = a(r n − 1)

Sn = a r

r

n( )−−

11

Month number Takings

1 $10 000

2 $ 8 500

3 $ 7 225

Year Height (m)

1 1.2

2 1.26

3 1.323

5F



This formula is useful if r < −1 or r > 1, for example, if r is 2, 10, 3.3, −4, −1.2.By calculating Sn − rSn instead of rSn − Sn, as we did earlier, we obtain an alternative form of

the formula. That is, Sn − rSn = a − ar n

Sn (1− r) = a (1 − r n)

Sn = a r

r

n( )11

−−

This formula is useful if r is in between −1 and 1 (shown as −1 < r < 1).

The sum of n terms, Sn, of a geometric sequence may be calculated using

Sa r

rn

n

= −−

( 1)1

if r < −1 or r > 1

or

Sa r

rn

n

= −−

(1 )1

if −1 < r < 1.

WoRked exAmple 16

Find the sum of the first 9 terms of the sequence 0.25, 0.5, 1, 2, 4, . . .

Think WRiTe/displAY

1 Find the value of a. a = 0.25

2 Find the value of r by testing ratios of the given terms.

tt2

1

0 50 25

= ..

tt3

2

10 5

=.

= 2 = 2tt4

3

21

=

tt5

4

42

=

= 2 = 2

3 Since r > 1, use Sa r

rn

n

= −−

( )11

. S9

90 25 2 11

= −. ( )

= 127.75

4 Write the answer. The sum of the first 9 terms is 127.75.

Alternatively, on the Main screen, complete the entry line as:

solve sa r n

rs= × −

−

( ^ ),

11

| a = 0.25

| r = 2 | n = 9Then press E.

230

The 3rd term of a geometric sequence is 11.25 and the 6th term is 303.75.Find the sum of the fi rst 10 terms of the sequence correct to 1 decimal place.

Think WRiTe

1 We need to fi nd a and r. We know that t3 = 11.25 and that tn = ar n − 1.

t3 = ar 2

= 11.25 [1]

2 We know that t6 = 303.75 and that tn = ar n − 1.

t6 = ar 5 = 303.75 [2]

3 Solve these equations simultaneously by eliminating a to fi nd r.

Equation [2] divided by equation [1]:

ar

ar

5

2

303 7511 25

= ..

r3 = 27 r = 3

4 Substitute r value into one of the equations to fi nd a, the fi rst term.

Substituting r = 3 into equation [1]: ar2 = 11.25a × 32 = 11.25

a = 11 25

9.

a = 1.25

5 Since r > 1, use Sa r

rn

n

= −−

( )11

. S10

101 25 3 12

= −. ( )

= 36 905

6 Write your answer. The sum of the fi rst ten terms of the geometric series is 36 905.

WoRked exAmple 18

How many terms of the geometric sequence 100, 95, 90.25, 85.7375, . . . are required for the sum to be greater than 1000?

Think WRiTe/displAY

Method 1: Using logarithms

1 Find a. a = 100

2 Find r.

r = =95100

0 95.

3 Use Sa r

rn

n

= −−

( )11

since r < 1.

Sn

n

= −100 1 0 950 05

( . ).

4 Investigate how many terms are required to sum to 1000 or Sn = 1000.

1000100 1 0 95

0 05= −( . )

.

n

= 2000(1 − 0.95n) 0.5 = 1 − 0.95n

0.95n = 0.5


Tutorialint-0449

Worked example 17



5 Express 0.95 and 0.5 in terms of logarithms with a base of 10. Of course,

other bases could be used. 131

2 terms are required to sum to 1000, so we require 14 to exceed 1000, as the number of terms is discrete.

log10 (0.95n) = log10 (0.5)n log10 (0.95) = log10 (0.5)

n =

log ( . )log ( . )

10

10

0 50 95

n = 13.513So, we require that n = 14.

Method 2: Using a CAS calculator1

2

3 The next whole number is 14. Write your answer.

The fi rst 14 terms are required for the sum to fi rst exceed 1000.

When the terms of a geometric sequence are added, a geometric series is formed:1. 2, 4, 8, 16, 32, 64 is a fi nite geometric sequence. 2 + 4 + 8 + 16 + 32 + 64 is a fi nite geometric series.

The sum of 2. n terms, Sn, of a geometric sequence may be calculated using

Sa r

rn

n

=−

( 1a r( 1a rn( 1n −( 1− )1

if r < −1 or r > 1 (for example, r = −2 or −3

2 or +2 or +1.5)

or

Sa r

rn

n

=a r−a r

−(1a r(1a r )1

if −1 < r < 1 (for example, r = 0.2 or 18 or −0.25).

RememBeR

To defi ne a geometric sum in the Main screen, tap:• Action• Command• Defi neComplete the entry line as:

Defi ne s na r n

r( )

( )= × −−

11

^ | a = 100 | r = 0.95

Then press E.

To solve for n when s(n) = 1000, tap:• Interactive• Advanced• solveComplete the equation box as:s(n) = 1000 Enter n in the variable box. Check ‘Solve numerically’ and then tap OK.

232

The sum of a given number of terms of a geometric sequence 1 a We 16 Find the sum of the first 12 terms of the geometric sequence 2, 6, 18, 54, 162, . . .

b Find the sum of the fi rst 7 terms of the geometric sequence 5, 35, 245, 1715, 12 005, . . .c Find the sum of the fi rst 11 terms of the geometric sequence 3.1, 9.3, 27.9, 83.7,

251.1, . . .d Find the sum of the fi rst 12 terms of the geometric sequence −0.1, −0.4, −1.6, −6.4,

−25.6, . . . 2 a We 17 The 2nd term of a geometric sequence is 10 and the 5th is 80. Find the sum of the

first 12 terms of the sequence.b The 2nd term of a geometric sequence is 15 and the 5th is 405. Find the sum of the fi rst

11 terms of the sequence.c The 2nd term of a geometric sequence is −12 and the 5th is −768. Find the sum of the fi rst

9 terms of the sequence.d The 3rd term of a geometric sequence is 500 and the 6th is −500 000. Find the sum of the

fi rst 10 terms of the sequence.

3 a We 18 How many terms of the geometric sequence 3, 6, 12, 24, 48, . . . are required for the sum to be greater than 3000?

b How many terms of the geometric sequence 5, 20, 80, 320, 1280, . . . are required for the sum to be greater than 100 000?

c How many terms of the geometric sequence 1.2, 2.4, 4.8, 9.6, 19.2, . . . are required for the sum to be greater than 10 000?

d How many terms of the geometric sequence 120, 96, 76.8, 61.44, 49.152, . . . are required for the sum to be greater than 540?

4 mC The sum of the first 10 terms of the geometric sequence 2.25, 4.5, 9, 18, 36, . . . is closest to:A 1149.75B 2301.75C 5318.81D 6648.51E 8342.65

5 mC The 2nd term of a geometric sequence is −20 and the 5th is −1280. The sum of the first 12 terms of the sequence is:A −27 962 025B −1 062 880C −15 360D 1 062 880E 16 777 215

6 Find the sum of the first 13 terms of the geometric sequence 80, 72, 64.8, 58.32, 52.488, . . . 7 Find the sum of the first 8 terms of the geometric sequence 250, −150, 90, −54, 32.4, . . . 8 Find, correct to 1 decimal place, the sum of the first 12 terms of the geometric sequence

−192, 48, −12, 3, −0.75. . . 9 The 3rd term of a geometric sequence is 2 and the 6th is 0.016. Find, correct to 1 decimal

place, the sum of the first 13 terms of the sequence.

10 The 3rd term of a geometric sequence is 90 and the 6th is 0.09. Find, correct to 1 decimal place, the sum of the first 14 terms of the sequence.

11 How many terms of the geometric sequence 600, 180, 54, 16.2, 4.86, . . . are required for the sum to be greater than 855?

exeRCise

5F

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Applications of geometric sequencesGrowth and decay of discrete variables is constantly found in real-life situations. Some examples are increasing or decreasing populations and increase or decrease in fi nancial investments. Some of these geometric models are presented here.

A city produced 100 tonnes of rubbish in the year 2004. Forecasts suggest that this may increase by 2% each year. If these forecasts are true:a what will be the city’s rubbish output in 2008?b in which year will the amount of rubbish reach 120 tonnes?c what was the total amount of rubbish produced by the city in the years 2004, 2005 and 2006?

This is an example of a geometric sequence where a = 100 and r = 1.02. Note that r ≠ 0.02. If this was the case, then multiplying 100 by 0.02 would result in a lesser amount of rubbish in the second year and so on. We are told that the amount of rubbish increases by 2%. That is the original amount plus an extra 2%, or: original amount + 2% of original amount = original amount (1 + 2%) = original amount (1 + 0.02) = 1.02 × original amount.

Think WRiTe/displAY

a 1 Find the fi rst term, a. a a = 100

2 Determine the common ratio, r. Increase by 2%1 + 2% = 1 + 0.02 r = 1.02

3 Determine which term is represented by the amount of rubbish for the year 2008.

Year 2004 is the fi rst term, so n = 1.Year 2005 is the second term, so n = 2.Year 2008 is the fi fth term, so n = 5.

4 Use tn = ar n − 1 to fi nd the amount of rubbish collected in the fi fth year.

t5 = 100 × 1.025 − 1

= 100 × 1.0824 = 108.24

5 Write your response. The amount of rubbish produced in the fi fth year, or 2008, will be 108.24 tonnes.


b 1 Use tn = ar n − 1 and tn = 120. b 100(1.02)n − 1 = 120 (1.02)n − 1 = 1.2

2 Express 1.02 and 1.2 in terms of logarithms with base 10.

log10 (1.02)n − 1 = log10 (1.2)(n − 1) log10 (1.02) = log10 (1.2)

n − 1 = log ( . )log ( . )

10

10

1 21 02

n − 1 = 9.207 n = 10.207

3 Write your answer. During the 11th year (that is, during 2014) the amount of rubbish produced will have exceeded 120 tonnes.

5G


Tutorialint-0450

Worked example 19

234


b 1 b

2 Write your answer. It is the 11th term or year when 120 tonnes is first exceeded; that is, 2014.

c 1 We need to find the sum of the first 3 years.

Use Sa r

rn

n

= −−

( )11

where n = 3.

c S3

3100 1 02 11 02 1

= −−

( . ).

= 306.04

2 Write your answer. The total output of rubbish for the years 2004, 2005 and 2006 was 306.04 tonnes.

Note: Alternatively, for part c , obtain the total amount by summing the first 3 values in the table on a CAS calculator from part b , or set up an equation using nSolve on a CAS calculator. Try these and check that you get the same solution.

WoRked exAmple 20

A computer decreases in value each year by 15% of the previous year’s value. Find an expression for the value of the computer, Vn, after n years. Its initial purchase price is given as V1 = $12 000.

Think WRiTe

1 This is a geometric sequence since there is a 15% decrease on the previous year’s value. Find a and r.Note: Since this is a decreasing value, r is a value less than 1.

a = 12 000 r = 1 − 15% = 1 − 0.15 = 0.85

2 We want an expression for the value after n years. Use tn = ar n − 1 which gives the value of the nth term. Use Vn instead of tn.

Vn = 12 000 × (0.85)n − 1

3 Write your answer. The value of the computer is given by the expression, Vn = 12 000(0.85)n − 1.

Compound interestConsider the case where a bank pays compound interest of 5% per annum on an amount of $20 000. The amount is invested for 4 years and interest is calculated yearly.

Compound interest receives its name because the interest which is earned is paid back into the account so that the next time interest is calculated, it is calculated on an increased amount. There is a compounding effect on the money in the account.


On the Spreadsheet screen, label column A as ‘n’ (for term number) and enter 1 in A2. In cell A3, complete the entry line as: = A2 + 1Label column B as ‘sum of’ and then complete the entry line in cell B2 as:= 100 × 1.02^(A2 − 1)Fill down both columns until column B exceeds 120.


If we calculated the amount in the account each year, we would have the following amounts.

Start $20 000After 1 year $20 000 × 1.05 = $21 000After 2 years $20 000 × 1.05 × 1.05 = $22 050After 3 years $20 000 × 1.05 × 1.05 × 1.05 = $23 152.50After 4 years $20 000 × 1.05 × 1.05 × 1.05 × 1.05 = $24 310.13

The amounts 20 000, 21 000, 22 050, 23 152.50, 24 310.13, . . . form a geometric sequence where a = 20 000 and r = 1.05.

We need to be a little careful, however, in using the formula tn = ar n − 1 in calculating compound interest. This is because the original amount in the account, that is, $20 000, in terms of the geometric sequence would be referred to as t1 or a. In banking terms, t1 would represent the amount in the account after the first lot of interest has been calculated and added in.

To be clear and to avoid errors, it is best to use the following formula for compound interest.

A = PR n

where R = 1 + r

100 A = amount in the account, $ P = principal, $ r = interest rate per period (e.g. per year, per quarter), % n = the number of periods during the investment.

Note: Students who are studying Module 4: Business related mathematics will use this formula for compound interest.

WoRked exAmple 21

Helen inherits $60 000 and invests it for 3 years in an account which pays compound interest of 8% per annum compounding each 6 months.a What will be the amount in Helen’s account at the end of 3 years?b How much will Helen receive in interest over the 3-year period?

Think WRiTe

a 1 This is an example of compound interest. Use A = PRn, where R

r= +1100

.

Interest is calculated each 6 months, so over 3 years, there are 6 periods: n = 6. Interest is 8% per year or 4% per 6 months.So, r = 4%.

a P = 60 000 n = 6 half-years r = 4% per half-year

So, R = +14

100 = 1.04 A = PRn

= 60 000(1.04)6

= 75 919.14

2 Write your answer. At the end of 3 years, Helen will have a total amount of $75 919.14.

b 1 Interest equals the amount in the account at the end of 3 years, less the amount in the account at the start of the investment.

b Interest = Total amount − Principal = $75 919.14 − $60 000 = $15 919.14

2 Write your answer. Amount of interest earned over 3 years is $15 919.14.

236

Jim invests $16 000 in a bank account which earns compound interest at the rate of 12% per annum compounding every quarter.

At the end of the investment, there is $25 616.52 in the account. For how many years did Jim have his money invested?

Think WRiTe/displAY


1 We know the value of A, P, r and R.We need to find n using the compound interest formula.Note: There are 4 quarters per annum.

A = 25 616.52P = 16 000 r = 12

4

= 3% per quarterand so R = 1 + 3

100 = 1.03

2 Now substitute into A = PRn. A = PRn

25 616.52 = 16 000(1.03)n

1.601 = 1.03n

3 Express 1.601 and 1.04 in terms of logarithms with base 10.

log10 (1.601) = log10 (1.03n)So, log10 (1.601) = n × log10 (1.03)

n =log ( . )

log ( . )10

10

1 601

1 03

n = 0 20440 0128..

4 Round up the number of periods to 16 to ensure the amount is reached.

n = 15.92

5 Write your answer. It will take 16 periods where a period is 3 months. So, it will take 48 months or 4 years.


1

2 Write your answer. The 16th term exceeds 25 616.25; hence it will take 16 periods where a period is 3 months. So, it will take 48 months or 4 years.

WoRked exAmple 22


On the Spreadsheet screen, label column A as ‘quarter’ and enter 1 in cell A2. In cell A3, complete the entry line as: = A2 + 1Label column B as ‘accountbalance’ and then complete the entry line in cell B2 as:= 16 000 × 1.03^(A2)Fill down both columns until column B exceeds $25 616.25.


Geometric growthGeometric growth or increase is expressed as a percentage increase.1. Common ratio, 2. r = 1 + % increaser3. values are greater than 1, for example, an 8% increase gives r = 1.08.

Geometric decayGeometric decay or decrease is expressed as a percentage decrease.4. Common ratio, 5. r = 1 − % decreaser6. values are less than 1, for example, an 8% decrease gives r = 0.92.

Compound interestThe formula for compound interest is7.

A = PRn

where R = 1 + r

100 A = amount in the account, $ P = principal, $ r = interest rate per period (e.g. per year, per quarter), % n = the number of periods during the investment.

RememBeR

Applications of geometric sequences 1 We 19 A farmer harvests 4 tonnes of lucerne in his first year of production. In his business

plan, he has estimated an annual increase of 6% on his lucerne harvest.a According to this plan, how many tonnes of lucerne should he harvest in his 7th year of

production?b In which year will his harvest reach

10 tonnes?c How much will he expect to harvest

in the fi rst three years?d How much in total will he harvest

from the 7th year to the 10th year?

2 A taxi driver estimates that the cost of keeping her taxi on the road increases by 4.5% each year. If the cost of keeping her taxi on the road in her first year of owning a taxi was $1800:a what was the cost in the 5th year?b during which year did costs exceed $2500?c what were the total costs of keeping her taxi on

the road in the fi rst 3 years?

3 We20 The population of a town is decreasing by 10% each year. Find an expression for the population of the town, which will be referred to as Pn. The population in the first year, P1, was 10 000.

4 We21 $13 000 is invested in an account which earns compound interest at 8% p. a. compounded quarterly (i.e. calculated four times each year).a After 5 years, how much is in the account?b How much interest was earned in that period?

exeRCise

5G

exAm Tip Students need to recognise that ‘from year 7 to year 10’ means that S S10 6− is required rather than S S10 7− .


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Digital docsSkillSHEET 5.5

Relating the common ratio of a geometric

sequence to percentage increase

or decreaseSpreadsheet 117

Sequences and series

238

5 The population of the newly established town of Alansford in its first year was 6000. It is predicted that the town’s population will increase by 10% each year. If this were to be the case, find:a the population of the town in its 10th yearb in which year the population of Alansford would reach 25 000.

6 The promoters of ‘Fleago’ flea powder assert that continued application of the powder will reduce the number of fleas on a dog by 15% each week. At the end of week 1, Fido the dog has 200 fleas left on him and his owner continues to apply the powder.a How many fl eas would Fido be expected to have on him at the end of the 4th week?b How many weeks would Fido have to wait before the number of fl eas on him had

dropped to less than 50?

7 Young saplings should increase in height by 9% each year under optimum conditions. If a batch of saplings which have been planted out measure 2.2 metres in their first year:a how high should they be in their 4th year?b in which year should they exceed 5 metres in

height?

8 mC A colony of ants is studied and the population of the colony in week 1 of the study is 800. If the population of the colony is expected to increase at the rate of 2% each week, then the week in which the number of ants would exceed 1000 would be closest to:A 6 B 10 C 13D 26 E 32

9 A company exported $300 000 worth of manufactured goods in its first year of production. According to the business plan of the company, this amount should increase each year by 7.5%.a How much would the company be expected to

export in its 5th year?b In which year would exports exceed $500 000?c What is the total amount exported by the company

in its fi rst 7 years of operation?

10 $10 000 is invested in an account which earns compound interest at 10% per annum. Find the amount in the account after 5 years if the interest is compounded:a yearly b every 6 monthsc quarterly d monthly.

11 $20 000 is invested in an account earning compound interest of 10% per annum compounding quarterly. What is the amount in the account after:a 1 year? b 3 years?c 5 years? d 10 years?

12 We22 In an account earning compound interest of 8% per annum compounding quarterly, an amount of $6000 is invested. When the account is closed, there is $7609.45 in the account. For how many years was the account open?

13 Sue earns 12% interest per annum compounding quarterly on her investment of $40 000. For how many years would this investment need to operate for the amount to rise to $50 670.80?

14 An amount of $14 500 is invested in an account attracting compound interest of 6% per annum compounding quarterly. After a certain time the interest earned in the account is $1834.14. Find out for how long the amount had been invested.



Finding the sum of an infinite geometric sequenceIf you are 2 metres away from a wall and you move 1 metre (or halfway) towards the wall and then move 1

2 metre (or halfway again) towards the wall and continue to do this, will you reach

the wall? When will you reach the wall?Consider the following geometric sequence:

1, 1

2, 1

4, 1

8, 1

16, . . . This is an infinite geometric sequence

since it continues on with an infinite number of terms. Each term in the sequence is less than the previous term by a factor of 1

2, that is, r = 0.5.

If we were to add n terms of this sequence together, we would have:

Sn

n

= × −−

1 1 0 51 0 5( . )

.

= −1 0 5

0 5.

.

n

= −1

0 50 50 5...

n

= 2 − 0.5n − 1

Consider 0.5n − 1 in the equation above. As n becomes very large, the term 0.5n − 1 becomes very small. Try this with your calculator.Let n = 5, 0.5n − 1 = 0.54 = 0.0625; therefore, S5 = 2 − 0.0625 = 1.9375Let n = 10, 0.5n − 1 = 0.59 = 0.001 95; therefore, S10 = 2 − 0.001 95 = 1.998 05Let n = 20, 0.5n − 1 = 0.519 = 0.000 001 9; therefore, S20 = 2 − 0.000 001 9 = 1.999 998 1

We can see that as n becomes larger, 0.5n − 1 becomes smaller. If n were to approach infinity (note that you can never reach infinity, you can only approach it), then the value of 0.5n − 1 would approach zero. So, Sn = 2 − 0.5n − 1 would become S∞ = 2.

It is possible to generalise this in order to find the sum of an infinite geometric sequence. We use the symbol S∞ which is referred to as the sum to infinity of a geometric sequence.

The sum to infinity of a geometric sequence for which −1 < r < 1 is given by

Sa

r∞∞ =−1

.

WoRked exAmple 23

Find the sum to infinity of the geometric sequence 2, 0.4, 0.08, 0.016, 0.0032, . . .

Think WRiTe

1 Find a and r. a = 2

r = tt2

1

= 0 42.

= 0.2

5h

1 m

1 m

m–21 –

41 –

81

2 m

240

2 As r = 0.2 satisfi es the condition−1 < r < 1, use the formula S

ar∞ =

−1.

Sa

r∞ =−1

= 2

1 0 2− .

= 2

0 8. = 2.5

3 Write your answer. The sum to infi nity of the given sequence is 2.5.

WoRked exAmple 24

The sum to infi nity of a geometric sequence is 15 and the value of a is 10. Write the fi rst 4 terms of the sequence.

Think WRiTe/displAY


1 Use the formula Sa

r∞ =−1

to fi nd the

value of r. Transpose the equation to

make r the subject.

Sa

r∞ =−1

15

101

=− r

1 10

15− =r

r = −1 2

3

r = 1

3


1

2 a = 10 and r = 13. Use these to generate

the terms in the sequence.

The fi rst 4 terms of the sequence are

10, 10

3, 10

9, 10

27 or 10, 31

3, 11

9, 10

27.


On the Main screen, tap:• Interactive• Advanced • solveEnter the equation in the equation box and r in the variable box. Then tap OK.Complete the entry line as:

solve sa

rr=

−

1

, | a = 10 | s = 15

Then press E.


The sum to infinity of the geometric sequence is 6.25 and the value of r is 0.2. Write the first 4 terms of the sequence.

Think WRiTe


r∞ =−1

to find the value of a. Sa

r∞ =−1

6 25

1 0 2.

.=

−a

6.25 × 0.8 = a a = 5

2 a = 5 and r = 0.2. Use these to generate the terms.

The first 4 terms of the sequence are 5, 1, 0.2, 0.04.

Converting recurring decimals to fractionsWe can use the sum to infinity formula to convert recurring decimals to fractions.

WoRked exAmple 26

Express 1.2. as a fraction.

Think WRiTe

1 We need to express 1 2.. as the sum of a geometric

sequence.1 2.

. = 1.222 222 222 222 2 . . .

= 1 + 0.2 + 0.02 + 0.002 + 0.0002 + . . . = 1 + (0.2 + 0.02 + 0.002 + 0.0002 + . . .)

2 The terms in the bracket form an infinite geometric sequence where a = 0.2 and r = 0.1. Use the formula.

Sa

r∞ =−1

.

Multiply both the numerator and the denominator

of 0 20 9..

by 10 to eliminate the decimal.

a = 0.2

r = 0 02

0 2..

= 0.1

S

ar∞ =

−1

00 2

1 0 12.

..

.=

−

00 20 9

2...

.=

029

2..

=

3 Express the final answer. So, 1 12 29

..

= +

= 1 2

9

WoRked exAmple 25

242

Express 0.645 . . . as a fraction.

Think WRiTe

1 We need to express 0.645 as the sum of a geometric sequence.

0.645 = 0.645 454 5 . . . = 0.6 + 0.045 + 0.000 45 + 0.000 004 5 + . . . = 0.6 + (0.045 + 0.000 45 + 0.000 004 5 + . . .)

2 The terms in the bracket form an infi nite geometric sequence where a = 0.045 and r = 0.01.

a = 0.045

r = 0.000 450.045

= 0.01


r∞ =−1

.

Multiply both the numerator and the

denominator of 0 0450 99..

by 1000 to

eliminate the decimal.

Sa

r∞ =−1

0 045

0 0451 0 01

..

.=

−

0 045

45990

. =

0 645 0 6

45990

. .= +

4 Express the fi nal answer.

= +610

45990

= +594

99045

990

5 Write your answer. So, 0 645639990

. =

WoRked exAmple 28

An injured rabbit attempts to crawl back to its burrow. It moves 30 metres in the fi rst hour, 21 metres in the second hour and 14.7 metres in the third hour and so on. If the burrow is 200 metres away, will the rabbit make it back?

Think WRiTe

1 Determine what sort of sequence we have.

t1 = 30, t2 = 21 and t3 = 14.7Now 21

30 = 0.7

and 14 7

21

. = 0.7So, we have a geometricsequence where a = 30, r = 0.7.

2 Find the value of Sa

r∞ =−1

. S∞ =−30

1 0 7. = 100

3 Write your answer. The rabbit will cover a total of 100 metres. Since the rabbit hole is 200 metres away, the rabbit won’t make it.


Tutorialint-0451

Worked example 27



For decreasing or decaying geometric series, the sum of an infi nite number of terms 1. approaches a fi nite sum.The sum to infi nity of a geometric sequence for which 2. −1 < r < 1 is given by

Sa

r∞ =−1

RememBeR

Finding the sum of an infi nite geometric sequence 1 We23 Find the sum to infinity of the following geometric sequences.

a 50, 25, 12.5, 6.25, 3.125, . . . b 20, 16, 12.8, 10.24, 9.192, . . .c 1, 1

3, 1

9, 1

27, 1

81, . . . d 1, 1

5, 1

25, 1

125, 1

625, . . .

e 3, −0.6, 0.12, −0.024, 0.0048, . . . f −50, 5, −0.5, 0.5, −0.05, . . .

2 We24 Write the first 3 terms of the geometric sequence for which:

a r = 0.6 and S∞ = 25 b r = 0.25 and S∞ = 8c r = 0.9 and S∞ = 120 d r = −0.2 and S∞ = 3

1

3

e r = −0.8 and S∞ = 5 f r = 0.2 and S∞ = −7.5

3 We25 Write the first 3 terms of the geometric sequence for which:a a = 12.5 and S∞ = 25 b a = 12.5 and S∞ = 50

c a = 48 and S∞ = 120 d a = 24

9 and S∞ = 32

3.

4 We26, 27 Express each of the following recurring decimals as fractions.

a 0 5. b 0 4..

c 1 3..

d 3 7..

e 8.666 666 666 . . . f 0 14.. .

g 0 529. h 1 321.

5 We28 A defiant child walks 10 metres towards his mother in the first minute, 4 metres in the second minute and 1.6 metres in the third minute. If the child continues to approach in this same pattern, and if his mother is standing stationary, 20 metres from the child’s initial position, will the child ever reach the mother?

6 A failing machine produces 35 metres of spouting in the first hour, 21 metres in the second hour and 12.6 in the third hour. If this pattern continues and 280 metres of spouting is required, how far short of the quota will the machine fall?

7 A nail penetrates 20 mm with the first hit of a hammer, 12 mm with the 2nd hit and 7.2 mm with the 3rd. If this pattern continues, will the 50 mm long nail ever be completely hammered in?

8 A woman establishes a committee to raise money for a hospital. It raises $40 000 in the first year, $36 800 in the 2nd year and $33 856 in the third year. If the fundraising continues in this pattern, how far short will they fall in raising $1 000 000?

exeRCise

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244

9 A will of a recently deceased woman specifies how her money is to be donated to a charity. Her total wealth of $12.5 million is to be donated for eternity with the first donation of $1 million in the first year.a What fraction of this fi rst donation should be donated for the second year and subsequent

years?b Write the value of the donations for each of the fi rst 5 years.c How much will be donated after 10 years?

Contrasting arithmetic and geometric sequences through graphsWhen discrete variables are presented graphically some distinct features may be evident. This is especially so for discrete variables that have an arithmetic or geometric pattern.

Arithmetic patternsArithmetic patterns are distinguished by a straight line or a constant increase or decrease.

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An increasing pattern or a positive common difference gives an upward straight line.

A decreasing pattern or a negative common difference gives a downward straight line.

Geometric patternsGeometric patterns are distinguished by a curved line or a saw form.

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An increasing pattern or a positive common ratio greater than 1 (r > 1) gives an upward curved line.

A decreasing pattern or a positive fractional common ratio (0 < r < 1) gives a downward curved line.

eBookpluseBookplus

Interactivityint-0186

Contrasting arithmetic and

geometric sequences

5i



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An increasing saw pattern occurs when the common ratio is a negative value less than −1 (r < −1).

A decreasing saw pattern occurs when the common ratio is a negative fraction (−1 < r < 0).

WoRked exAmple 29

On the graph at right, the first 5 terms of a sequence are plotted. State whether the sequence could be arithmetic or geometric and give the value of a and the value of either d or r.

Think WRiTe

1 Examine the difference between the value of each of the terms. In each case, they are the same, that is, 10.

There is a constant difference between each successive term so the straight line graph shows an arithmetic sequence.

2 Find a. Now, t1 = 40, so a = 40.

3 Find d. Now, t2 = 30 and t1 = 40.So, d = t2 − t1 = −10.

WoRked exAmple 30

Calculate the total amount in an account if $10 000 is invested for 5 years and earns:a simple interest of 10% per annum

b compound interest of 10% per annum compounding yearly.

c For each of the above cases, graph, on the same set of axes, the amount in the account over the five years. Use your graph or calculations to calculate the difference between the accounts after 4 years.

Think WRiTe/displAY

a Calculate how much is in the account earning simple interest at the end of each of the five years.

a After 1 year, amount in account = 10 000 + 10% of 10 000 = 10 000 + 1000 = $11 000After 2 years, amount in account = 10 000 + 2 × 10% of 10 000 = $12 000After 3 years = $13 000After 4 years = $14 000After 5 years = $15 000

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b Calculate the amount in the account earning compound interest at the end of each of the 5 years using A = PRn where R = 1 + r

100

(that is, 1 + 10% = 1.1).

b After 1 year, amount in account = 10 000 × 1.11

= $11 000After 2 years, amount in account = 10 000 × 1.12

= $12 100After 3 years = $13 310After 4 years = $14 641After 5 years = $16 105

Method 1: Plotting the graph

c 1 Draw the graphs of the amount in the account earning simple interest (straight line) and the amount in the account earning compound interest (curved line) on the same set of axes.

c

Number of years invested (n)3 4 5210

16 00015 000

13 00012 00011 00010 000

14 000

17 000

Am

ount

in a

ccou

nt (

$) Compound interest

Simple interest

2 Use the values calculated for the end of the fourth year.

Difference in amounts = 14 641 − 14 000 = $641

3 Write your answer. After 4 years, the compound interest account earns an extra $641 in interest.


c 1 c


On the Spreadsheet screen, label column A as ‘s-year’ and enter 1 into cell A2. For cell A3, complete the entry line as:= A1 + 1Use the stylus to highlight the column down to the 6th year. Then tap:• Edit• Fill Range

Note: Since the first term, a, is also known as t1, let n = 1 represent the start of first year so that:n = 1 start of 1st year Initial depositn = 2 start of 2nd year end of 1st year. . .n = 6 start of 6th year end of 5th year


2

3

4

5 As defined earlier, ‘after 4 years’ is the same as the beginning of the 5th year. That is, n = 5. Write your answer.

After 4 years the difference is $641.

Label column B as simpbal. The rule is:tn = 10 000 + (n − 1) × 1000 wherea = 10 000 is the first term and d = 1000 is the common difference; that is, 10% of $10 000.In cell B2, complete the entry line as:= 10 000 + (A2 − 1) × 1000

Label column C as ‘compbal’. The rule is: tn = 10 000 × 1.1n − 1 where a = 10 000 is the first term and r = 1.1 is the common ratio; that is, 10% increase is 1 + 0.1 or 1.1.In cell C2, complete the entry line as:=10 000 × 1.1^(A2 − 1)Fill down both columns until the 6th year.

To view a graph comparing simple and compound interest, open the Spreadsheet screen and highlight the columns ‘s-year’, ‘simpbal’ and ‘compbal’. Tap:• Graph• Scatter Ensure ‘Column Series’ is checked. Then tap: • View• Zoom Box • r

248

Arithmetic patterns1.

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Geometric patterns2.

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RememBeR



Contrasting arithmetic and geometric sequences through graphs 1 We29 On the graph at right, the first five terms of a

sequence are plotted. State whether the sequence could be arithmetic or geometric and give the value of a and the value of either d or r.

2 On the graph at right, the first five terms of a sequence are plotted. State whether the sequence could be arithmetic or geometric and give the value of a and the value of either d or r.



5 Draw a graph showing the first 8 terms of each of the following sequences:a arithmetic, a = 7, d = 2b geometric, a = 5, r = 1

2c arithmetic, a = −14, d = 3.5d arithmetic, a = 32, d = −5e geometric, a = 12, r = 10

3

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6 mC On the graph below, the first five terms of a sequence are plotted.

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The sequence could be described by which one of the following?A Arithmetic sequence with a = 10 and d = 10B Arithmetic sequence with a = 10 and d = 0.5C Geometric sequence with a = 10 and r = 0.5D Geometric sequence with a = 10 and r = 2E Geometric sequence with a = 10 and r = 1.5

7 mC On the graph below, the first five terms of a sequence are plotted.

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The sequence could be described by which one of the following?A Arithmetic sequence with a = 10 and d = 20B Arithmetic sequence with a = 10 and d = −20C Geometric sequence with a = 10 and r = 10D Geometric sequence with a = 10 and r = −20E Geometric sequence with a = 10 and r = −1

8 We30 An amount of $5000 is invested for 3 years and earns:a simple interest of 10% per annumb compound interest of 10% per annum compounding yearly.On the same set of axes, plot points showing the amount in each account at the end of each of the 3 years.

9 An amount of $100 000 is invested for 3 years and earns:a simple interest of 15% per annumb compound interest of 15% per annum compounding yearly.On the same set of axes, plot points showing the amount in each account at the end of each of the 3 years.

10 On the same set of axes, sketch the graphs of the sequences with the rule un = 10n and vn = 10 × 1.5n − 1. Use your graph to decide for how many of the first five terms un is greater than vn.

11 On the same set of axes, sketch the graphs of the sequences with the rule un = 120 − 20n and vn = 100 × 0.8n − 1. Use your graph to decide for how many of the first five terms un is greater than vn.



summARY

Recognition of arithmetic sequences

An arithmetic sequence is a sequence of numbers for which the difference between successive terms is the same.•Given an arithmetic sequence, identify:• the first term, a and the common difference, d = t2 − t1 .Given an unspecified sequence, establish whether it is arithmetic by testing all terms for a common •difference: d = t2 − t1 = t3 − t2 = t4 − t3 = . . .

Finding the terms of an arithmetic sequence

t• n = a + (n − 1) d where tn is the nth term. a is the first term d is the common difference

The sum of a given number of terms of an arithmetic sequence

A • series is the sum of terms in a sequence.S• n is the sum of the first n terms in series, for example, S25 represents the sum of the first 25 terms.Given a number of terms in a series, • n, first term, a and the last term, l, use

Sn

a ln = +2

( ).

Given a number of terms in a series, • n, first term, a and the common difference, d, use

Sn

a n dn = + −2

2 1[ ( ) ].

Recognition of geometric sequences

A geometric sequence is a sequence of numbers for which the ratio of successive terms is the same.•Given a geometric sequence, identify: • the first term, a and the common ratio, r =

t

t2

1

.

Given an unspecified sequence, establish whether it is geometric by testing all terms for a common ratio, •

r = tt

tt

tt

2

1

3

2

4

3= = = . . .

Finding the terms of a geometric sequence

t• n = ar n − 1

where tn is the nth term a is the first term r is the common ratio.

The sum of a given number of terms of a geometric sequence

2, 4, 8, 6, 32, 64 is a finite geometric sequence.•2 • + 4 + 8 + 16 + 32 + 64 is a finite geometric series.The sum of • n terms, Sn, of a geometric sequence may be calculated using

Sn = a r

r

n( )−−

11

if r > 1 or r < −1 (for example r = −2 or − 32

or +2 or +4.5).

or

Sn = a r

r

n( )11

−−

if −1 < r < 1 (for example, r = 0.2 or 18 or −0.25).

252

Geometric growth

Growth or increase is expressed as a percentage increase.•Common ratio, • r = 1 + % increaser• values are greater than 1, for example, an 8% increase gives r = 1.08.

Geometric decay

Decay or decrease is expressed as a percentage decrease.•Common ratio, • r = 1 − % decreaser• values are less than 1, for example, an 8% decrease gives r = 0.92.

Compound interest

A• = PRn where R = 1 + r100

and A = amount in the account, $ P = principal, $ r = interest rate per period (e.g. per year, per quarter), % n = the number of periods during the investment.

Finding the sum of an infinite geometric sequence

For decreasing or decaying geometric series, the sum of an infinite number of terms approaches a finite sum.•The sum to infinity of a geometric sequence for which • −1 < r < 1 is given by

S∞ = a

r1−.

Contrasting arithmetic and geometric sequences through graphs

Arithmetic patterns are distinguished by a straight line.•

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Geometric patterns are distinguished by a curved line or a saw form.•

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254

ChApTeR RevieW

mulTiple ChoiCe

1 Which of the following could be the first 5 terms of an arithmetic sequence?A 1, 2, 4, 8, 12, . . .B 3, 3, 6, 6, 9, . . .C 56, 57, 58, 59, 60, . . .D −5, 5, 10, 15, 20, . . .E 1, 4, 9, 16, 25, . . .

2 For the sequence −3.6, −2.1, −0.6, 0.9, 2.4, . . ., it is true to say it is:A an infinite sequence with a = −3.6 and d = −0.15B an infinite sequence with a = −3.6 and d = 1.5C an infinite sequence with a = −0.15 and d = −3.6D a finite sequence with a = −0.15 and d = −3.6E a finite sequence with a = −3.6 and d = 0.15.

3 For the arithmetic sequence, −1, 1, 3, 5, 7, . . . the value of a, the value of d and the rule for the sequence are given respectively by:A a = −1, d = 2, tn = −3 + 2nB a = −1, d = 2, tn = −3 − nC a = 1, d = −1, tn = 2 − nD a = 2, d = −1, tn = 3 − nE a = 2, d = −1, tn = 3 − n

4 The 43rd term of the arithmetic sequence −7, 2, 11, 20, 29, . . . is:A −327 B −243 C 371D 380 E 387

5 The 3rd term of an arithmetic sequence is 3.1 and the 7th term is −1.3. The value of the 31st term is:A −153.7 B −27.7 C 28.9D 38.3 E 157.9

6 The sum of the first 24 terms of the sequence −16, −12, −8, −4, 0, . . . is:A 720 B 912 C 1344D 1440 E 1488

7 The first term of an arithmetic sequence is 14 and the 3rd is 8. The sum of the first 30 terms of the sequence is:A −1770 B −1095 C −885D 1725 E 2190

8 There is a geometric sequence for which a = 3 and r is a negative number. We can be certain that:A r is a fraction less than 1B the 3rd term will be a positive numberC the 3rd term will be greater than the 1st term

D only one number in the sequence is positiveE the 4th term will be greater than the 3rd term

9 Which of the following is a geometric sequence?A 2, −2, 2, −2, 2, . . .B 2, 4, 6, 8, 10, . . .C 1, 1

3,

− 19

, 127

, − 181

, . . .

D 4, −4, 2, −2, 1, . . .E 100, 10, 0.1, 0.01, 0.001, . . .

10 The 19th term of the geometric sequence 3.25, 6.5, 13, 26, 52, . . . is:A 425 984 B 851 968 C 1 703 936D 41 978 243 E 3 272 883 098

11 The 3rd term of a geometric sequence is 19.35 and the 6th is 522.45. The 12th term of the sequence is:A 16 539.15 B 417 629.75 C 126 955.35D 380 866.05 E 1 142 598.15

12 The first 3 terms of a geometric sequence are 2.25, 4.5, 9. The first term to exceed 1000 is:A t9 B t10 C t11D t12 E t13

13 The sum of the first 10 terms of the geometric sequence 8, 4, 2, 1, 1

2, . . . is closest to:

A 15 B 16 C 17D 18 E 20

14 The 3rd term of a geometric sequence is 0.9 and the 6th is 7.2. The sum of the first 12 terms of the sequence is closest to:A −2 B 122 C 921D 4122 E 8190

15 A tree increases in height each year by 5%. If it was 1.2 m high in its first year, in its 6th year its height would be closest to:A 1.53 m B 1.61 m C 5.5 mD 9.11 m E 3750 m

16 Profits in a company are projected to increase by 8% each year. If the profit in the first year was $60 000, in which year could a profit in excess of $100 000 be expected?A year 6 B year 7 C year 8D year 9 E year 10

17 The sum to infinity of the geometric sequence 1, 4

51625

64125

256625

, , , , . . . is:

A 15

B 54

C 145

D 4 E 5



18 The first term of the geometric sequence for which r = −0.5 and S∞ = 5

13 is:

A −1 B −213 C 2

23

D 8 E 1023

19 The first five terms of a sequence are plotted on the graph at right.

The sequence could be described by which of the following?A Arithmetic sequence with a = 50 and d = 25B Arithmetic sequence with a = 50 and d = 0.5C Geometric sequence with a = 50 and r = 0.5D Geometric sequence with a = 50 and r = 1.5E Geometric sequence with a = 50 and r = 2

20 The first five terms of a sequence are plotted on the graph below.

The sequence could be described by which of the following?A Arithmetic sequence with a = 10 and d = −5B Arithmetic sequence with a = 10 and d = −0.5C Geometric sequence with a = 10 and r = 5D Geometric sequence with a = 10 and r = −5E Geometric sequence with a = 10 and r = 0.5

21 A crystal measures 12.0 cm in length at the beginning of a chemistry experiment. Each day it increases in length by 3%. The length of the crystal after 14 days’ growth is closest to:A 12.4 cm B 16.7 cm C 17.0 cmD 17.6 cm E 18.2 cm [VCAA 2006]

22 A healthy eating and gym program is designed to help football recruits build body weight over an extended period of time. Roh, a new recruit who initially weighs 73.4 kg, decides to follow the program.

In the fi rst week he gains 400 g in body weight.In the second week he gains 380 g in body weight.In the third week he gains 361 g in body weight.If Roh continues to follow this program indefi nitely, and this pattern of weight gain remains the same, his eventual body weight will be closest to:A 74.5 kg B 77.1 kg C 77.3 kgD 80.0 kg E 81.4 kg [VCAA 2006]

shoRT AnsWeR

1 For the sequences below, state whether or not they are an arithmetic sequence. If they are, give the value of a and d.a −123, −23, 77, 177, 277, . . .b −51

4, −21

4, 3

4, 33

4, 63

4, . . .

2 If the second term of an arithmetic sequence is −5 and the fifth term is 16, which term in the sequence is equal to 226?

3 Blood donations at a suburban location increase by 40 each year. If there are 520 donations in the first year:a how many donations are made in the

15th year?b what is the total number of donations made

over those 15 years?

4 For each of the sequences below, state whether or not they are a geometric sequence. If they are, state the value of a and r.

a 5, 52, 5

4, 5

8, 5

16, . . .

b −700, −70, −7, 7, 70, . . . 5 The amount of garbage (in tonnes) collected in a

particular area by the local council each year is recorded over 3 successive years.

Year number Amount of garbage (tonnes)

1 7.2

2 8.28

3 9.522

If the amount collected each year were to continue to follow a geometric sequence:

a write a rule for the amount of garbage, tn, which would be collected in the area in year n

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b how much garbage would be collected in the 8th year? (Answer correct to 2 decimal places.)

c in which year would the amount of garbage collected exceed 30 tonnes?

exAm Tip A difference equation was not appropriate here as a formula in terms of n was required.


6 How many terms of the geometric sequence 164, 131.2, 104.96, 83.968, 67.1744, . . . are required for the sum to exceed 800?

7 Andrew invests $25 000 in an account earning compound interest of 10% per annum compounding quarterly.

a Find the amount in the account after 3 years.b Find how long it would take to have $40 965.41

in his account.

8 Express 3 7. as a fraction.

9 The batteries in a toy soldier are running down. The toy soldier marches 50 cm in the first minute, 30 cm in the second minute, 18 cm in the next and so on. By how much does the toy soldier fall short of marching 1.5 m?

10 On the same set of axes, sketch the graph of the sequence with the rule:a un = 10nb vn = 10 × 2n − 1.

exTended Response

Task 1

1 Consider the geometric sequence 1, 3, 9, . . . , whose common ratio is 3.a Subtract successive terms to form the sequence 2, 6, . . . Is this a geometric sequence as well and, if so,

what is its common ratio?b Add successive terms to form the sequence 4, 12, . . . Is this a geometric sequence as well and, if so, what

is its common ratio?c Repeat parts a and b for multiplication and division.d Prove your result of part a for any geometric sequence.

2 In this problem we are comparing simple and compound interest.Consider a bank, which offers a simple interest rate of 5% per annum on an investment of $100.a What is the value of the investment for each of the

first 5 years?b Consider another bank, which offers compound

interest at the same rate of 5%. What is the value of the investment for each of the first 5 years?

c When will the value of the investment in part b be twice as much as the investment in part a?

Task 2

1 A newly established quarry produces crushed rock for the building of roads and freeways. The amount of crushed rock (in tonnes) it produces increases by 3

12 tonnes each month and its production for the first

3 months of operation is shown below.

Month Crushed rock produced (tonnes)

1 8

2 11.5

3 15

a Write the amount of crushed rock produced in the 4th month.b Write a rule for tn, the amount of crushed rock produced in month n, expressed in terms of n, the

nth month.c Write the amount of crushed rock produced in the 60th month.

exAm Tip Students need to be able to show a clear method for a calculation (or table) where a term in an arithmetic sequence first exceeds a term of a geometric sequence.




d During which month will the amount of crushed rock coming from the quarry exceed 100 tonnes?e The local council has ordered that after a total of 3050 tonnes of crushed rock has been extracted from

the quarry, an environmental impact survey must be completed. After how many months will that happen?

2 The amount of crushed rock produced each month at a second quarry is shown below.

Month Crushed rock produced (tonnes)

1 10

2 11

3 12.1

Given that production at this quarry increases geometrically, fi nd:a the common ratio, rb a rule for the amount of crushed rock produced, tn, in tonnes, expressed in terms of the number of

months, nc the amount of crushed rock produced in the 5th monthd in which month the amount of crushed rock produced exceeds 30 tonnese the total amount of crushed rock produced by the quarry in its fi rst year of operation.

3 During its first month of production, the second quarry produces more crushed rock than the first quarry. In the months after that, however, the first quarry produced more crushed rock than the second quarry. After how many months does the second quarry produce more than the fi rst quarry again?

Task 3A new type of fruit tree is planted in the orchard. In the fi rst month after planting, the gardeners worked 625 hours. In the second and third months after planting, the gardeners worked 500 hours and 400 hours respectively. Suppose this decreasing pattern of work continues and the number of hours the gardeners work each month follows a geometric sequence.a Show that the common ratio of this sequence is r = 0.8.b Determine the number of hours the gardeners will work in the fi fth month.c Write an expression that gives the number of hours, Hn, the gardeners will work in the nth month after

planting.d How many more hours will the gardeners work in the sixth month than in the seventh month? Write your

answer correct to the nearest hour.e In which month will the gardeners fi rst work less than 100 hours? In the fi rst three months after planting, the gardeners worked a total of 1525 hours.f How many hours, in total, will the gardeners work in the next nine months? Write your answer correct to the

nearest hour. [VCAA 2006]

eBookpluseBookplus

Digital docTest Yourself

Chapter 5

258

eBookpluseBookplus ACTiviTies

Chapter openerDigital doc

10 Quick Questions: Warm up with a quick quiz on •arithmetic and geometric sequences. (page 202)

5B Finding the terms of an arithmeticsequence

Digital docs

SkillSHEET 5.1: Practise solving simultaneous •equations using the elimination method. (page 211)SkillSHEET 5.2: Practise solving linear equations •(arithmetic terms). (page 211)

Tutorial

We 5 • int-0446: Watch a worked example on fi nding the value of a term in an arithmetic sequence given its value and the beginning of an arithmetic sequence. (page 209)

Interactivity int-0007Number patterns: Recognise the relationship •between two variables by observing patterns. (page 207)

5C The sum of a given number of terms of an arithmetic sequence

Digital doc

WorkSHEET 5.1: Recognise arithmetic sequences •and series. (page 218)

Tutorial

We 8 • int-0447: Watch a tutorial on fi nding the sum of an arithmetic sequence using the formula. (page 215)

5E Finding the terms of a geometric sequence Digital docs

SkillSHEET 5.3: Practise solving non-linear •simultaneous equations (page 227)SkillSHEET 5.4: Practise solving indicial equations •(page 227)

Tutorial

We 14 • int-0448: Learn how to fi nd the value of a term in a geometric sequence using a CAS calculator and by using a spreadsheet. (page 225)

5F The sum of a given number of terms of a geometric sequence

Digital doc

WorkSHEET 5.2: Recognise geometric sequences •and series. (page 232)

Tutorial

We 17 • int-0449: Watch a tutorial on fi nding the value of the sum of a geometric sequence given two non-consecutive terms. (page 230)

5G Applications of geometric sequencesDigital docs

SkillSHEET 5.5: Practise relating the common ratio •of a geometric sequence to percentage increase or decrease. (page 237)Spreadsheet 117: Investigate graphs of arithmetic •sequences and series. (page 237)

Tutorial

We 19 • int-0450: Watch a worked example on applying the concepts involved in geometric sequences to real life. (page 233)

5H Finding the sum of an infinite geometricsequence

Tutorial

We 27 • int-0451: Watch a worked example on expressing a recurring decimal as a fraction using the sum to infi nity formula. (page 242)

5I Contrasting arithmetic and geometric sequences through graphs

Interactivity int-0186Contrasting arithmetic and geometric sequences: •Consolidate your understanding of arithmetic and geometric sequences. (page 244)

Chapter review Digital doc

Test Yourself: Take the end-of-chapter test to test •your progress. (page 257)

To access eBookPLUS activities, log on to

www.jacplus.com.au


Documents

5A 5B 5C 5D Arithmetic 5E 5F and geometric 5G 5H sequences 5Imathsbooks.net/JACPlus Books/12 Further Maths/Ch05... · 1. An arithmetic sequence is a sequence of numbers for which