Geometric Sequences & Geometric Sequences & SeriesSeries
8.38.3
JMerrill, 2007JMerrill, 2007
Revised 2008Revised 2008
SequencesSequences
A Sequence:A Sequence:
Usually defined to be a functionUsually defined to be a function
Domain is the set of positive integersDomain is the set of positive integers
Arithmetic 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
Is the Sequence 3, 8, 13, 18…Is the Sequence 3, 8, 13, 18…
A.A. ArithmeticArithmetic
B.B. GeometricGeometric
C.C. NeitherNeither
Is the Sequence 2, 5, 10, 17…Is the Sequence 2, 5, 10, 17…
A.A. ArithmeticArithmetic
B.B. GeometricGeometric
C.C. NeitherNeither
Is the Sequence 8, 12, 18, 27…Is the Sequence 8, 12, 18, 27…
A.A. ArithmeticArithmetic
B.B. GeometricGeometric
C.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 2
12a
r1 5
96a
r
ExampleExample
1 2
12a
r1 5
96a
r
2 5
5 2
3
12 96
12 96
8
2
r rr r
r
r
1
( 1
1
1
1
)1
101
123
4*
3*2
3072
nn
a
a a r
a
a
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
n
a rS
r
Notice – 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
10
2(1 - (-3) ) 2(1 - 3 )S = =
1- -3 29,524
4
1(1 )1
n
n
a 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 2
n
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)
1
1
a
Sr
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
1
4
1
114
11 314
aS
r
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
1
1 2 3
1 2 4 7...
S
S
S
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
aS
r
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
aS
r
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; 0.01
0.45 5
1 1 0.01 11
a r
aS
r
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; 0.1
0.7 7
1 1 0.1 9
a r
tS
r
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
2
1
11
2
3
1
16 ; 241
121
24 122
112 6
2
tS
rt
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 1
6 2
10
n
12(1 2 )S 12,276
1 2
12
24
2