12.1 An Introduction to 12.1 An Introduction to Sequences & SeriesSequences & Series

SequenceSequence::• A list of ordered numbers separated by A list of ordered numbers separated by

commas. commas. • Each number in the list is called a Each number in the list is called a termterm..• For Example:For Example:

Sequence 1Sequence 1 Sequence 2Sequence 2 2,4,6,8,102,4,6,8,10 2,4,6,8,10,… 2,4,6,8,10,…

Term 1, 2, 3, 4, 5Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5Term 1, 2, 3, 4, 5DomainDomain – relative position of each term (1,2,3,4,5) – relative position of each term (1,2,3,4,5)

Usually begins with position 1 unless otherwise Usually begins with position 1 unless otherwise stated.stated.

RangeRange – the actual terms of the sequence – the actual terms of the sequence (2,4,6,8,10)(2,4,6,8,10)

Sequence 1Sequence 1 Sequence 2Sequence 2

2,4,6,8,102,4,6,8,10 2,4,6,8,10,…2,4,6,8,10,…

A sequence can be A sequence can be finitefinite or or infiniteinfinite..

The sequence has The sequence has a last term or a last term or finalfinal

term.term.

(such as seq. 1)(such as seq. 1)

The sequence The sequence continues without continues without

stopping.stopping.

(such as seq. 2)(such as seq. 2)Both sequences have a Both sequences have a general rulegeneral rule: a: ann = 2n where = 2n where

n is the term # and an is the term # and ann is the nth term. is the nth term.

The general rule can also be written in The general rule can also be written in function function notationnotation: f(n) = 2n: f(n) = 2n

Examples:Examples:• Write the first 6 Write the first 6

terms of aterms of ann=5-n.=5-n.

• aa11=5-1=4=5-1=4

• aa22=5-2=3=5-2=3

• aa33=5-3=2=5-3=2

• aa44=5-4=1=5-4=1

• aa55=5-5=0=5-5=0

• aa66=5-6=-1=5-6=-1

• 4,3,2,1,0,-14,3,2,1,0,-1

• Write the first 6 Write the first 6 terms of aterms of ann=2=2nn..

• aa11=2=211=2=2

• aa22=2=222=4=4

• aa33=2=233=8=8

• aa44=2=244=16=16

• aa55=2=255=32=32

• aa66=2=266=64=64

• 2,4,8,16,32,642,4,8,16,32,64

ExamplesExamples: Write a rule for the nth term.: Write a rule for the nth term.

The seq. can be The seq. can be written as:written as:

Or, aOr, ann=2/(5=2/(5nn))

• The seq. can be The seq. can be written as:written as:

2(1)+1, 2(2)+1, 2(3)+1, 2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,…2(4)+1,…

Or, aOr, ann=2n+1=2n+1

,...625

2,

125

2,

25

2,

5

2 .a

,...5

2,

5

2,

5

2,

5

24321

,...9,7,5,3 .b

Example: write a rule for the nth term.

• 2,6,12,20,…2,6,12,20,…

• Can be written as:Can be written as:

1(2), 2(3), 3(4), 4(5),…1(2), 2(3), 3(4), 4(5),…

Or, aOr, ann=n(n+1)=n(n+1)

Graphing a SequenceGraphing a Sequence• Think of a sequence as ordered pairs for Think of a sequence as ordered pairs for

graphing. (n , agraphing. (n , ann))

• For example: 3,6,9,12,15 For example: 3,6,9,12,15 would be the ordered pairs (1,3), (2,6), would be the ordered pairs (1,3), (2,6), (3,9), (4,12), (5,15) graphed like points in a (3,9), (4,12), (5,15) graphed like points in a scatter plotscatter plot

* Sometimes it helps to find the rule first * Sometimes it helps to find the rule first when you are not given every term in a when you are not given every term in a finite sequence.finite sequence.

Term #Term # Actual termActual term

Graphing

na

1

3

2

6

3

9

4

12

• What is a sequence?A collections of objects that is ordered so that

there is a 1st, 2nd, 3rd,… member.• What is the difference between finite and

infinite?

Finite means there is a last term. Infinite means the sequence continues without stopping.

Sequences and Series• What is a series?• How do you know the difference between a How do you know the difference between a

sequence and a series?sequence and a series?

• What is sigma notation?• How do you write a series with summation

notation?• Name 3 formulas for special series.

SeriesSeries• The sum of the terms in a sequence.The sum of the terms in a sequence.

• Can be finite or infiniteCan be finite or infinite

• For Example:For Example:

Finite Seq.Finite Seq. Infinite Seq.Infinite Seq.

2,4,6,8,102,4,6,8,10 2,4,6,8,10,…2,4,6,8,10,…

Finite SeriesFinite Series Infinite SeriesInfinite Series

2+4+6+8+102+4+6+8+10 2+4+6+8+10+…2+4+6+8+10+…

Summation NotationSummation Notation• Also called Also called sigma notationsigma notation

(sigma is a Greek letter (sigma is a Greek letter ΣΣ meaning “sum”) meaning “sum”)

The series 2+4+6+8+10 can be written as:The series 2+4+6+8+10 can be written as:

i is called the i is called the index of summationindex of summation

(it’s just like the n used earlier). (it’s just like the n used earlier).

Sometimes you will see an n or k here instead of i.Sometimes you will see an n or k here instead of i.

The notation is read:The notation is read:

““the sum from i=1 to 5 of 2i”the sum from i=1 to 5 of 2i”

5

1

2ii goes from 1 i goes from 1

to 5.to 5.

Summation Notation for an Summation Notation for an Infinite SeriesInfinite Series

• Summation notation for the infinite series:Summation notation for the infinite series:

2+4+6+8+10+… would be written as:2+4+6+8+10+… would be written as:

Because the series is infinite, you must use i Because the series is infinite, you must use i from 1 to infinity (from 1 to infinity (∞) instead of stopping at ∞) instead of stopping at

the 5the 5thth term like before. term like before.

1

2i

Examples: Write each series in Examples: Write each series in summation notation.summation notation.

a. 4+8+12+…+100a. 4+8+12+…+100• Notice the series can Notice the series can

be written as:be written as:

4(1)+4(2)+4(3)+…+4(25)4(1)+4(2)+4(3)+…+4(25)

Or 4(i) where i goes Or 4(i) where i goes from 1 to 25.from 1 to 25.

• Notice the series Notice the series can be written as:can be written as:

25

1

4i

...5

4

4

3

3

2

2

1 . b

...14

4

13

3

12

2

11

1

. to1 from goes where1

Or,

ii

i

1 1i

i

ExampleExample: Find the sum of the : Find the sum of the series.series.

• k goes from 5 to 10.k goes from 5 to 10.

• (5(522+1)+(6+1)+(622+1)+(7+1)+(722+1)+(8+1)+(822+1)+(9+1)+(922+1)+(10+1)+(1022+1)+1)

= 26+37+50+65+82+101= 26+37+50+65+82+101

= = 361361

10

5

2 1k

Special Formulas (shortcuts!)Special Formulas (shortcuts!)

nn

i

1

12

)1(

1

nni

n

i

6

)12)(1(

1

2

nnni

n

i

1

n

i

c cn

Example: Find the sum.Example: Find the sum.

• Use the 3Use the 3rdrd shortcut! shortcut!

10

1

2

i

i

6

)12)(1( nnn

6

)110*2)(110(10

6

21*11*10 385

6

2310

• What is a series?A series occurs when the terms of a sequence are

added.• How do you know the difference between a

sequence and a series?The plus signs• What is sigma notation?∑• How do you write a series with summation

notation?Use the sigma notation with the pattern rule.• Name 3 formulas for special series.

1

n

i

c cn

2

)1(

1

nni

n

i 6

)12)(1(

1

2

nnni

n

i