Sec 11.3Geometric Sequences and Series

Objectives:•To define geometric sequences and series.•To define infinite series.•To understand the formulas for sums of finite and infinite geometric series.

An arithmetic sequence is defined when we repeatedly add a number, d, to an initial term.

A geometric sequence is generated when we start with a number, a1 , and repeatedly multiply that number by a fixed nonzero constant, r, called the common ratio.

Definition of a Geometric Sequence

A geometric sequence is a sequence of the forma, ar, ar2, ar3, ar4, . . The number a is the first

term, and r is the common ratio of the sequence.

The nth term of a geometric sequence is given by 1n

na ar

Ex. 1 Find the first four terms and the nth term of the geometric sequence with a = 2 and r = 3.

Since a = 2 and r = 3, we can plug into the nth term formula to get the terms.

1

01

12

23

34

2(3) 2

2(3) 6

2(3) 18

2(3) 54

nna ar

a

a

a

a

Ex 2. Find the eighth term of the geometric sequence 5, 15, 45,…

1

78 5(3)

5(2187)

10,935

nna ar

a

Ex 3. The third term of a geometric sequence is 63/4, and the sixth term is 1701/32. Find the fifth term.

6

3

1701

3263

4

a

a

5

2

1701

3263

4

ar

ar

Using the nth term formula and the two given terms we get a system of equations.

If we divide the equations we get the following:

3

170132634

r

And so, we get: 3 27

83

2

r

r

Since we have the sixth term, just divide it by r and you will get the fifth term.

6

5

5

170156732

3 162

aa

r

a

Partial Sums

For the geometric sequence

a, ar, ar2, ar3, ar4, . . . , arn–1, . . . ,

the nth partial sum is:

1

1

n

n

rS a

r

Ex 4. Find the sum of the first five terms of the geometric sequence 1, 0.7, 0.49, 0.343, . . .

5

5

5

1

1

1 0.71

1 0.7

2.7731

n

n

rS a

r

S

S

Ex. 5 Find the sum.

5

23

1

7k

k

5

5

1

1

21

14 323

13

n

n

rS a

r

S

Plug 1 in for k and you will get a = -14/3. Then plug in a and r into the sum formula.

5

32114 243

533

27514 243

533

770

243

S

You do not want to use decimals. All answers will be exact (which means they will be given as fractions.) We do not want to get rounded answers.

Use the fraction key on your calculator to help you get the numbers you see here.

HW #3 Finite Geometric Series Wkst odds (even extra credit)

Infinite Series

An expression of the form a1 + a2 + a3 + a4 + . . .

is called an infinite series.

Let’s take a look at the partial sums of this series.

1 1 1 1 1

2 4 8 16 2n

Sum of an Infinite Geometric Series

If | r | < 1, the infinite geometric series a + ar + ar2 + ar3 + ar4 + . . . + arn–1 + . . .

has the sum

1

aS

r

Ex 6 Find the sum of the infinite geometric series

2 2 2 22

5 25 125 5n

HW #4 Infinite Series Wkst odds(evens extra credit)