11.4 Geometric Sequences

11.4 Geometric Sequences and Series

If we start with a number, a1, and repeatedly multiply it by some constant, r, then we have a geometric geometric sequencesequence:

a1, a1r, a1r 2, a1r

3, a1r 4,….

The nth term of a geometric sequence is given by

The number r is called the common ratio

11 1, for any integern

na a r n

11.4 Geometric Sequences and Series (Example 2)

Find the 8th term of the geometric sequence: 5, 15, 45, …

Solution:

Use formula, an = a1r (n – 1)

a1 = r = n =

(Now work Example 3 in text….)

Geometric Series The sum of the terms of a geometric sequence

is called a geometric seriesgeometric series. For example:

is a finite geometric series with common ratio r.

What is the sum of the first n terms of a finite geometric series?

2 3 2 1.... n nnS a ar ar ar ar ar

Deriving a formula for the nth partial sum of a geometric sequence

2 2 1.... n nnS a ar ar ar ar

2 2 1 + .... + n n nnrS ar ar ar ar ar

0 0 0 0 . . . . . nn nS rS a ar

1 1( ) ( )nnS r a r

1

1

( )

( )

n

n

rS a

r

11.4 Geometric Sequences and Series The sum of the terms of an infinite

geometric sequence is an infinite infinite geometric seriesgeometric series. For some geometric sequences, Sn gets

close to a specific number as n gets large. This number becomes the limit of the sum

of the infinite geometric sequence. When |r|<1, the limit or sum of an infinite

geometric series is given by .1

1

aS

r

11.4 Geometric Sequences and Series

You should be able to: Identify the common ratio of a

geometric sequence, and find a given term and the sum of the first n terms.

Find the sum of an infinite geometric series, if it exists.

11.4 Sequences and Series

You should be able to: Find terms of sequences given the nth

term. Convert between sigma notation and

other notation for a series.