MATHPOWERTM 12, WESTERN EDITION

6.5

6.5.1

Chapter 6 Sequences and Series

6.5.2

Geometric Series

A geometric series is the sum of a geometric sequence.The formula for a geometric series is:

Sn a(r n 1)

r 1

Example: Find the sum of the series 5 + 15 + 45 + . . . + 10 935.

Sn a(r n 1)

r 1

Sn 5(38 1)

3 1

Sn = 16 400

tn = arn - 1

10 935 = 5(3)n - 1

2187 = 3n - 1

37 = 3n - 1

7 = n - 1 8 = n

The sum of the series is 16 400.

6.5.3

Find the sum of the first seven terms of the series 27 + 9 + 3 + . . .:

Geometric Series

Sn a(r n 1)

r 1

S7 27

1

3

7

1

1

3 1

S7 27

2186

2187

2

3

S7 2186

54

The sum of the first seven terms is1093

27or approximately 40.5.

6.5.4

Geometric Series

How many terms of the series 2 + (-4) + 8 + (-16) + . . .will yield a sum of 342?

Sn a(r n 1)

r 1

342 2(( 2)n 1)

2 1

-1026 = 2((-2)n - 1) - 513 = (-2)n - 1 -512 = (-2)n

(-2)9 = (-2)n

9 = n For this geometric series, t9 = 342.Therefore, the sum of the first nineterms is 342.

6.5.5

Applications --The Bouncing Ball

A ball is dropped from a height of 100 m and bounces back to 40% of its previous height. Find the height ofthe ball after it hits the floor for the fourth time.

tn = arn - 1

= 100(0.40)4

= 2.56 mThe vertical height of the ballafter the fourth bounce is 2.56 m.

6.5.6

The Bouncing Ball [cont’d]Find the total vertical distance travelled by the ballwhen it contacts the floor for the fifth time.

100 m

The total vertical distancetravelled is the sum of theupward and downward distances.

The total vertical distance will be 2Sn - 100.

Sn a(r n 1)

r 1

S5 100((0.4)5 1)

0.4 1

S5 = 164.96

Stotal = 2(164.96) - 100 = 229.92

The total vertical distancetravelled is 229.92 m.

6.5.7

Applications--The Telephone Fan-Out

Student Student

Student

Student Student

Student

Teacher Level 1

Level 2

Level 3

a) How many students will be contacted at the 8th level?b) At what level will 64 students be contacted?c) By the 8th, how many students will be contacted altogether?d) By the nth level, how many students will be contacted altogether?e) Suppose there are 300 students to be contacted. By what level will all have been contacted?

20

21

22

6.5.8

The Telephone Fan-Out [cont’d]

a) How many students will be contacted at the 8th level?

e) Suppose there are 300 students to be contacted. By what level will all have been contacted?

28 - 1 = 27 or 128 students

b) At what level will 64 students be contacted?

2n - 1 = 64 n = 7

at the 7th level

c) By the 8th level, how many students will be contacted altogether?

S8 1(28 1)

2 1S8 = 255 254 students

d) By the nth level, how many students will be contacted altogether? Sn = 2n - 1

by the 9th level

N.B. 255 =254 students+ 1 teacher

6.5.9

Using Sigma Notation

Write the following series using sigma notation and thenfind the sum of the series:27 + 81 + 243 + 729 + 2187 + 6561

tn = arn - 1

= 27(3)n - 1

= (33)(3n - 1) = 3n + 2

3k 2

k1

6

Summation notation for

this series is:

Sn a(r n 1)

r 1

S6 27(36 1)

3 1

S6 = 9828 The sum of the series is 9828.

6.5.10

Suggested Questions:Pages 309 and 3101-21 odd, 22,23, 28, 32 a