An Introduction to Sequences & Series p. 651. Sequence: A list of ordered numbers separated by...

An Introduction to An Introduction to Sequences & SeriesSequences & Series

p. 651

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

term.term.

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

The sequence continues The sequence continues without stopping.without 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 n is

the term # and athe 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 notationfunction notation: : f(n) = 2nf(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

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

2i

i 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

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

Arithmetic Sequences Arithmetic Sequences & Series& Series

Arithmetic Sequence:Arithmetic Sequence:

• The difference between consecutive The difference between consecutive terms is constant (or the same).terms is constant (or the same).

• The constant difference is also known The constant difference is also known as the as the common difference common difference (d).(d).

(It’s also that number that you are adding (It’s also that number that you are adding everytime!)everytime!)

Example: Decide whether each Example: Decide whether each sequence is arithmetic.sequence is arithmetic.

• -10,-6,-2,0,2,6,10,…-10,-6,-2,0,2,6,10,…

• -6--10=4-6--10=4

• -2--6=4-2--6=4

• 0--2=20--2=2

• 2-0=22-0=2

• 6-2=46-2=4

• 10-6=410-6=4

Not arithmetic (because Not arithmetic (because the differences are the differences are not the same)not the same)

• 5,11,17,23,29,…5,11,17,23,29,…

• 11-5=611-5=6

• 17-11=617-11=6

• 23-17=623-17=6

• 29-23=629-23=6

• Arithmetic (commonArithmetic (common difference is 6)difference is 6)

Rule for an Arithmetic SequenceRule for an Arithmetic Sequence

aann=a=a11+(n-1)d+(n-1)d

ExampleExample:: Write a rule for the nth Write a rule for the nth term of the sequence 32,47,62,77,… . term of the sequence 32,47,62,77,… .

Then, find a Then, find a1212..

• The is a common difference where d=15, The is a common difference where d=15, therefore the sequence is arithmetic.therefore the sequence is arithmetic.

• Use aUse ann=a=a11+(n-1)d+(n-1)d

aann=32+(n-1)(15) =32+(n-1)(15)

aann=32+15n-15=32+15n-15

aann=17+15n=17+15n

aa1212=17+15(12)=197=17+15(12)=197

ExampleExample: One term of an arithmetic sequence : One term of an arithmetic sequence is ais a88=50. The common difference is 0.25. =50. The common difference is 0.25.

Write a rule for the nth term.Write a rule for the nth term.• Use aUse ann=a=a11+(n-1)d to find the 1+(n-1)d to find the 1stst term! term!

aa88=a=a11+(8-1)(.25)+(8-1)(.25)

50=a50=a11+(7)(.25)+(7)(.25)

50=a50=a11+1.75+1.75

48.25=a48.25=a11

* Now, use a* Now, use ann=a=a11+(n-1)d to find the rule.+(n-1)d to find the rule.

aann=48.25+(n-1)(.25)=48.25+(n-1)(.25)

aann=48.25+.25n-.25=48.25+.25n-.25

aann=48+.25n=48+.25n

Now graph an=48+.25n.

• Just like yesterday, remember to graph the Just like yesterday, remember to graph the ordered pairs of the form (n,aordered pairs of the form (n,ann))

• So, graph the points (1,48.25), (2,48.5), So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc. (3,48.75), (4,49), etc.

Example: Two terms of an arithmetic sequence are Example: Two terms of an arithmetic sequence are aa55=10 and a=10 and a3030=110. Write a rule for the nth term.=110. Write a rule for the nth term.

• Begin by writing 2 equations; one for each term Begin by writing 2 equations; one for each term given.given.

aa55=a=a11+(5-1)d OR 10=a+(5-1)d OR 10=a11+4d+4d

AndAnd

aa3030=a=a11+(30-1)d OR 110=a+(30-1)d OR 110=a11+29d+29d• Now use the 2 equations to solve for aNow use the 2 equations to solve for a11 & d. & d.

10=a10=a11+4d+4d

110=a110=a11+29d (subtract the equations to cancel a+29d (subtract the equations to cancel a11))

-100= -25d -100= -25d

So, d=4 and aSo, d=4 and a11=-6 (now find the rule)=-6 (now find the rule)

aann=a=a11+(n-1)d+(n-1)d

aann=-6+(n-1)(4) OR a=-6+(n-1)(4) OR ann=-10+4n=-10+4n

Example (part 2):Example (part 2): using the rule a using the rule ann=-10+4n, =-10+4n,

write the value of n for which awrite the value of n for which ann=-2.=-2.

-2=-10+4n-2=-10+4n

8=4n8=4n

2=n2=n

Arithmetic SeriesArithmetic Series• The sum of the The sum of the

terms in an terms in an arithmetic sequencearithmetic sequence

• The formula to find The formula to find the sum of a finite the sum of a finite arithmetic series is:arithmetic series is:

2

1 nn

aanS

# of terms# of terms

11stst Term Term

Last Last TermTerm

ExampleExample: Consider the arithmetic : Consider the arithmetic series 20+18+16+14+… .series 20+18+16+14+… .

• Find the sum of the 1Find the sum of the 1stst 25 terms.25 terms.

• First find the rule for First find the rule for the nth term.the nth term.

• aann=22-2n=22-2n

• So, aSo, a2525 = -28 (last term) = -28 (last term)

• Find n such that SFind n such that Snn=-760=-760

2

1 nn

aanS

2

28202525S 100)4(2525 S

2

1 nn

aanS

2

)222(20760

nn

-1520=n(20+22-2n)-1520=n(20+22-2n)

-1520=-2n-1520=-2n22+42n+42n

2n2n22-42n-1520=0-42n-1520=0

nn22-21n-760=0-21n-760=0

(n-40)(n+19)=0(n-40)(n+19)=0

n=40 or n=-19n=40 or n=-19

Always choose the positive solution!Always choose the positive solution!

2

)222(20760

nn

Geometric Sequences & Geometric Sequences & SeriesSeries

Geometric SequenceGeometric Sequence

• The ratio of a term to it’s previous term The ratio of a term to it’s previous term is constant.is constant.

• This means you multiply by the same This means you multiply by the same number to get each term.number to get each term.

• This number that you multiply by is This number that you multiply by is called the called the common ratiocommon ratio (r). (r).

ExampleExample: Decide whether each : Decide whether each sequence is geometric.sequence is geometric.

• 4,-8,16,-32,…

• -8/4=-2

• 16/-8=-2

• -32/16=-2

• Geometric (common ratio is -2)

• 3,9,-27,-81,243,…

• 9/3=3

• -27/9=-3

• -81/-27=3

• 243/-81=-3

• Not geometric

Rule for a Geometric SequenceRule for a Geometric Sequence

aann=a=a11rrn-1n-1

ExampleExample: Write a rule for the nth term of the : Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,… . Then find asequence 5, 2, 0.8, 0.32,… . Then find a88..

•First, find r.First, find r.

•r= r= 22//5 5 = .4= .4

•aann=5(.4)=5(.4)n-1n-1

aa88=5(.4)=5(.4)8-18-1

aa88=5(.4)=5(.4)77

aa88=5(.0016384)=5(.0016384)

aa88=.008192=.008192

ExampleExample: One term of a geometric sequence : One term of a geometric sequence is ais a44=3. The common ratio is r=3. Write a rule =3. The common ratio is r=3. Write a rule

for the nth term. Then graph the sequence.for the nth term. Then graph the sequence.

• If aIf a44=3, then when n=4, =3, then when n=4,

aann=3.=3.

• Use aUse ann=a=a11rrn-1n-1

3=a3=a11(3)(3)4-14-1

3=a3=a11(3)(3)33

3=a3=a11(27)(27)11//99=a=a11

• aann=a=a11rrn-1n-1

aann=(=(11//99)(3))(3)n-1n-1

• To graph, graph the To graph, graph the points of the form points of the form (n,a(n,ann).).

• Such as, (1,Such as, (1,11//99), ),

(2,(2,11//33), (3,1), (4,3),…), (3,1), (4,3),…

Example: Two terms of a geometric sequence are Example: Two terms of a geometric sequence are aa22=-4 and a=-4 and a66=-1024. Write a rule for the nth term.=-1024. Write a rule for the nth term.

• Write 2 equations, one for each given term.

a2=a1r2-1 OR -4=a1r

a6=a1r6-1 OR -1024=a1r5

• Use these 2 equations & substitution to solve for a1 & r.

-4/r=a1

-1024=(-4/r)r5

-1024=-4r4

256=r4

4=r & -4=r

If r=4, then a1=-1.

an=(-1)(4)n-1

If r=-4, then a1=1.

an=(1)(-4)n-1

an=(-4)n-1

Both Both Work!Work!

Formula for the Sum of a Finite Formula for the Sum of a Finite Geometric SeriesGeometric Series

r

raS

n

n 1

11

n = # of termsn = # of terms

aa1 1 = 1= 1stst term term

r = common ratior = common ratio

Example: Consider the geometric Example: Consider the geometric series 4+2+1+½+… .series 4+2+1+½+… .

• Find the sum of the first 10 terms.

• Find n such that Sn=31/4.

r

raS

n

n 1

11

21

1

21

14

10

10S

128

1023

1024

20464

21

10241023

4

211024

11

410

S

21

1

21

14

4

31

n

21

1

21

14

4

31

n

2121

14

4

31

n

n

2

118

4

31

n

2

11

32

31n

2

1

32

1n

2

1

32

1

5n

n

n

2

1

32

1

n232 log232=n

p.675

Infinite Geometric SeriesInfinite Geometric Series

The sum of an The sum of an infinite geometric infinite geometric

seriesseries

1r if , 1

1

r

aS

sum. no is there,1 If r

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

1

1)1.0(2i

i

For this series, aFor this series, a11=2 & r=0.1=2 & r=0.1

1.1

2

S

9

20

9.

2

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

9

4

3

4412 So, a1=12

and r=1/3

31

1

12

S

32

12S

2

36S

S=18

Example: An infinite geom. Series has aExample: An infinite geom. Series has a11=4 & =4 &

a sum of 10. What is the common ratio?a sum of 10. What is the common ratio?

r

aSuse

1 1

r

1

410

10(1-r)=4

1-r = 2/5

-r = -3/5

5

3r

Example: Write 0.181818… as a Example: Write 0.181818… as a fraction.fraction.

0.181818…=18(.01)+18(.01)0.181818…=18(.01)+18(.01)22+18(.01)+18(.01)33+…+…

Now use the rule for the sum!Now use the rule for the sum!

r

a

11

01.1

18.

99.

18.

11

2