6.3 Geometric Series

3/25/2013

Geometric Sequence:

Is a sequence where each term is multiplied by the same factor in order to obtain the following term.

Example: 2, 8, 32, 128, 512, . . .

4

x 4 x 4 x 4

Common ratio (r):

x 4

3

Rule of nth term of a geometric sequence

2, 8, 32, 128, 512, . . .

Common ratio, r = 4

1st term: 2nd term: 3rd term: 4th term: What’s the pattern?Rule:

Geometric Series:

Expression formed by adding the terms of a geometric sequence.

Example(Finite): ∑

𝑛=0

8

6 (4 )𝑛

GeometricInfinite Series:

∑𝑛=0

∞

7( 12 )𝑛

6¿

7 ( 12 )0

+7 ( 12 )1

+7 ( 12 )2

+…. .

𝑎1r𝑎1

r

a. Yesb. Each term is multiplied by 5.c. r = 5

a. Determine whether the following series is geometric.b. Explain why or why not.c. If it is geometric, find the common ratio.

a. Yesb. Each term is multiplied by .c. r = .

a. Nob. Each term is multiplied by different values.

a. Nob. Each term is multiplied by different values.

Sum of the Finite Geometric Series

S = Sum of the series = first term in the series.

r = common ration = number of terms.

Find the sum of the finite geometric series

+ +

r = 4n = since n started at 0 there are 9 terms

= 524,286

Find the sum of the finite geometric series

r = n = 15

= 13.999957

Sum of the Infinite Geometric Series

Only when common ratio is between 0 and 1.

S = Sum of the series

= first term in the series.r = common ratio

Find the sum of the infinite geometric series

r = = 14

Find the sum of the infinite geometric series

r = =

Homework

WS 6.3 odd problems only

“I stayed up all night to see where the sun went. Then it dawned on me.”