Geometric Sequences & Series 8.3

Geometric Sequences & Geometric Sequences & SeriesSeries

8.38.3

JMerrill, 2007JMerrill, 2007Revised 2008Revised 2008

SequencesSequences

A Sequence:A Sequence:Usually defined to be a functionUsually defined to be a functionDomain is the set of positive integersDomain is the set of positive integersArithmetic sequence graphs are linear Arithmetic sequence graphs are linear (usually)(usually)Geometric sequence graphs are Geometric sequence graphs are exponentialexponential

Geometric SequencesGeometric Sequences

GEOMETRICGEOMETRIC - the ratio of any two consecutive - the ratio of any two consecutive terms in constant.terms in constant.Always take a number and divide by the Always take a number and divide by the preceding number to get the ratiopreceding number to get the ratio 1,3,9,27,81……….1,3,9,27,81……….

ratio = 3ratio = 364,-32,16,-8,4……64,-32,16,-8,4……

ratio = -1/2ratio = -1/2a,ar,ara,ar,ar22,ar,ar33………………

ratio = rratio = r

What is the ratio of 4, 8, 16, 32…What is the ratio of 4, 8, 16, 32…

22

What is the ratio of 27, -18, 12,-8…What is the ratio of 27, -18, 12,-8…

-2/3-2/3

Is the Sequence 3, 8, 13, 18…Is the Sequence 3, 8, 13, 18…

A.A. ArithmeticArithmeticB.B. GeometricGeometricC.C. NeitherNeither





ExampleExample

Write the first six terms of the geometric Write the first six terms of the geometric sequence with first term 6 and common sequence with first term 6 and common ratio 1/3.ratio 1/3.

2 2 2 26,2, , , ,3 9 27 81

Formulas for the nFormulas for the nthth term of a term of a SequenceSequence

Geometric:Geometric: aann == aa11 * * r r (n(n-1)-1)

To get the To get the nnthth term term, start with the , start with the 11stst term term and multiply by the and multiply by the ratioratio raised to the raised to the (n-1)(n-1) powerpower

n = THE TERM NUMBER

ExampleExample

Find a formula for aFind a formula for ann and sketch the graph and sketch the graph for the sequence 8, 4, 2, 1...for the sequence 8, 4, 2, 1...

Arithmetic or Geometric? Arithmetic or Geometric? r = ? r = ? aann = a = a1 1 ((r r ((nn-1)-1) ) )aann = 8 * ½ = 8 * ½ ((nn-1)-1)

12

n = THE TERM NUMBER

Using the FormulaUsing the Formula

Find the 8Find the 8thth term of the geometric term of the geometric sequence whose first term is -4 and whose sequence whose first term is -4 and whose common ratio is -2common ratio is -2

aann == aa11 * * r r (n(n-1)-1)

aa88 == -4-4 * * (-2) (-2) (8(8-1)-1)

aa88 = -4(-128) = 512 = -4(-128) = 512

ExampleExample

Find the given term of the geometric Find the given term of the geometric sequence if asequence if a33 = 12, a = 12, a66 =96, find a =96, find a1111

r = ? Since ar = ? Since a1 1 is unknown. Use given infois unknown. Use given info

aann = = aa11 * r * r ((nn-1)-1) aann = = aa11 * r * r ((nn-1)-1)

aa33 = = aa11 * r * r22 aa66 = = aa11 * r * r55

12 12 = a = a11 *r *r22 96 = a96 = a11 *r *r55

1 212ar

1 596ar

ExampleExample

1 212ar

1 596ar

2 5

5 2

3

12 96

12 9682

r rr r

rr

1

( 1

1

1

1

)1

101

12 34*

3*23072

nn

a

a a r

aa

Sum of a Finite Geometric SeriesSum of a Finite Geometric Series

The sum of the first n terms of a geometric The sum of the first n terms of a geometric series isseries is

1(1 )1

n

na rS

rNotice – no last term needed!!!!

ExampleExample

Find the sum of the 1Find the sum of the 1stst 10 terms of the 10 terms of the geometric sequence: 2 ,-6, 18, -54geometric sequence: 2 ,-6, 18, -54

10 10

102(1 - (-3) ) 2(1 - 3 )S = =

1- -3 29,5244

1(1 )1

n

na rS

rWhat is n? What is a1? What is r?

That’s It!

Infinite Geometric SeriesInfinite Geometric Series

Consider the infinite Consider the infinite geometric sequencegeometric sequence

What happens to each What happens to each term in the series?term in the series?They get smaller and They get smaller and smaller, but how small smaller, but how small does a term actually does a term actually get?get?

1 1 1 1 1, , , ,... ...2 4 8 16 2n

Each term approaches 0

Partial SumsPartial Sums

Look at the sequence of partial sums:Look at the sequence of partial sums:

1

2

3

121 1 32 4 41 1 1 72 4 8 8

S

S

S

What is happening to the sum?

It is approaching 1

0

1

Here’s the RuleHere’s the Rule

Sum of an Infinite Geometric SeriesSum of an Infinite Geometric Series If |r| < 1, the infinite geometric seriesIf |r| < 1, the infinite geometric series

aa11 + a + a11r + ar + a11rr22 + … + a + … + a11rrn n + …+ …

converges to the sumconverges to the sum

If |r| > 1, then the series diverges (does not have a If |r| > 1, then the series diverges (does not have a sum)sum)

11aSr

Converging – Has a SumConverging – Has a Sum

So, if -1 < r < 1, then So, if -1 < r < 1, then the series will the series will converge. Look at converge. Look at the series given bythe series given by

Since r = , we know Since r = , we know that the sum that the sum

is is

The graph confirms: The graph confirms:

1 1 1 1 ...4 16 64 256

14

1

114

11 31 4

aSr

Diverging – Has NO SumDiverging – Has NO Sum

If r > 1, the series will If r > 1, the series will diverge. Look at diverge. Look at 1 + 2 + 4 + 8 + …. 1 + 2 + 4 + 8 + …. Since r = 2, we know Since r = 2, we know that the series grows that the series grows without bound and without bound and has no sum. has no sum.

The graph confirms: The graph confirms:

1

2

3

11 2 31 2 4 7...

SSS

ExampleExample

Find the sum of the infinite geometric Find the sum of the infinite geometric series 9 – 6 + 4 - …series 9 – 6 + 4 - …

We know: aWe know: a11 = 9 and r = ? = 9 and r = ?2

3

1 9 2721 51

3

aSr

You TryYou Try

Find the sum of the infinite geometric Find the sum of the infinite geometric series 24 – 12 + 6 – 3 + …series 24 – 12 + 6 – 3 + …

Since r = -½ Since r = -½ 1

124 24 48 161 3 31

2 2

aSr

S

ExampleExample

Ex: The infinite, repeating decimal Ex: The infinite, repeating decimal 0.454545… can be written as the infinite 0.454545… can be written as the infinite seriesseries

0.45 + 0.0045 + 0.000045 + …0.45 + 0.0045 + 0.000045 + …What is the sum of the series? (Express What is the sum of the series? (Express the decimal as a fraction in lowest terms)the decimal as a fraction in lowest terms)

1

1

0.45; 0.010.45 5

1 1 0.01 11

a raSr

You TryYou Try

Express the repeating decimal, 0.777…, Express the repeating decimal, 0.777…, as a rational number (hint: the sum!)as a rational number (hint: the sum!)

1

1

0.7; 0.10.7 7

1 1 0.1 9

a rtSr

You Try, Part DeuxYou Try, Part Deux

Find the first three terms of an infinite Find the first three terms of an infinite geometric sequence with sum 16 and geometric sequence with sum 16 and common ratio common ratio 1

21

11

2

3

1

16 ; 24112124 12

2112 6

2

tSrt t

t

t

Last ExampleLast Example

Find the following sum:Find the following sum:

What’s the first term? What’s the first term? What’s the second term? What’s the second term? Arithmetic or Geometric?Arithmetic or Geometric?What’s the common ratio?What’s the common ratio?Plug into the formula…Plug into the formula…

10i

i 16 2

10

n12(1 2 )S 12,2761 2

12

24

2

Can You Do It???Can You Do It???

Find the sum, if possible, of Find the sum, if possible, of

88

0

142

n

n

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Geometric Sequences & Series 8.3