Geometric Sequences

Definition of a Geometric Sequence

• A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.

The Common Ratio

The common ratio, r, is found by dividing any term after the first term by the term that directly precedes it. In the following examples, the common ratio is found by dividing the second term by the first term, a2 a1.

 Geometric sequence Common ratio1, 5 25, 125, 625... r 5 1 = 56, -12, 24, -48, 96.... r -12 6 -2

Write the first six terms of the geometric sequence with first term 6 and common ratio 1/3 .

Solution The first term is 6. The second term is 6 1/3, or 2. The third term is 21/3, or2/3.The fourth term is 2/31/3, or2/9, and so on. The first six terms are

6, 2,2/3,2/9,2/27,2/81.

Text Example

General Term of a Geometric Sequence

• The nth term (the general term) of a geometric sequence with the first term a1 and common ratio r is

• an = a1 r n-1

Find the eighth term of the geometric sequence whose first term is 4 and whose common ratio is 2.

Solution To find the eighth term, a8, we replace n in the formula with 8, a1 with 4, and r with 2.

an a1r n 1

a88 1 4)7 4(128)

The eighth term is 512. We can check this result by writing the first eight terms of the sequence:

4, 8, 16, 32, 64, 128, 256, 512.

Text Example

The Sum of the First n Terms of a Geometric Sequence

r

raS

n

n

1

)1(1

The sum, Sn, of the first n terms of a geometric sequence is given by

in which a1 is the first term and r is the common ratio.

Example

5314404

)5314411(4

)3(1

))3(1(4

1

)1(

1

)1(

12121

12

1

r

raS

r

raS

n

n

• Find the sum of the first 12 terms of the geometric sequence: 4, -12, 36, -108, ...Solution:

Value of an Annuity: Interest Compounded n Times per YearIf P is the deposit made at the end of each

compounding period for an annuity at r percent annual interest compounded n times per year,

the value, A, of the annuity after t years:

nr

nr

PA

nt 1)1(

Example• To save for retirement, you decide to deposit

$2000 into an IRA at the end of each year for the next 30 years. If the interest rate is 9% per year compounded annually, find the value of the IRA after 30 years.

Solution: P=2000, r =.09, t = 30, n=1

109.

1)109.

1(2000

1)1( 30*1

nr

nr

PA

nt

Example cont.• To save for retirement, you decide to deposit

$2000 into an IRA at the end of each year for the next 30 years. If the interest rate is 9% per year compounded annually, find the value of the IRA after 30 years.

Solution:

08.615,272$09.

2677.122000

09.

1)09.1(2000

1

09.

1)1

09.1(

2000

30

30*1

The Sum of an Infinite Geometric Series

If -1<r<1, then the sum of the infinite geometric series

a1+a1r+a1r2+a1r3+…

in which a1 is the first term and r s the common ration is given by

r

aS

11

If |r|>1, the infinite series does not have a sum.

Geometric Sequences