CN College Algebra Ch. 11: Sequences 11.3: Geometric Sequences Goals: Determine if a sequence is geometric.

Find a formula for a geometric sequence.

Find the sum of a geometric sequence.

Find the sum of a geometric series.

Geometric Sequences:



Geometric Sequence

: When the

ratio

of successive terms of a sequence is always the same nonzero number, the sequence is called geometric. A geometric sequence may be defined recursively as

a

1 

a

,

a n



ar n

 1 , where a = a 1 , and r ≠ 0 are real numbers. The number a is the first term, and the nonzero number r is called the

common ratio

.

Geometric Sequence Theorems:



Theorem – n th Term of a Geometric Sequence

: For a geometric sequence {a n } whose first term is a and whose common ratio is r, the n th term is determined by the formula

a n



ar n

 1 ,

r

 0 .



Theorem – Sum of n Terms of a Geometric Sequence

: Let {a n } be a geometric sequence with first term a and common ratio r, where r ≠ 0, r ≠ 1. The sum S n of the first n terms of {a n } is

S n



a

1  1 

r n r

,

r

 0 ,

r

 1 .

Geometric Series:



Geometric Series

: An infinite sum of the form

a



ar



ar

2   

ar n

  with first term a and common ratio r, is called an infinite  geometric series and is denoted by

ar k

 1

k

  1 

Sum of an infinite geometric series

: The sum S n

n

first n terms of a geometric series is

S n



a

1 1  

r r

 1

a



r

 of the 1

ar



n r

.

If this finite sum Sn approaches a number L as n   , then we call L the sum of the infinite geometric series  and we write

L



ar k

 1 .

k

  1

Sum of an Infinite Geometric Series:



Theorem – Sum of an Infinite Geometric Series

: If |r| < 1, the sum of the infinite geometric series is

k

   1

ar k

 1 

a

1 

r

.

k

   1

ar k

 1

Assignments:

 Class work: Sequences and Series worksheet.

 Homework: 11.3 Exercises #1-5 (odds), 11-21 (odds), 25-29 (odds), 33, 39, 41, 71.