Geometric Sequences and Series

Geometric Sequences

Find the general term of the following:

1. 1, 2, 4, 8, 16, ...

2. 27, 9, 3, 1, 1/3, ...

3. 3, 6, 12, 24, 48, ...

4. 1/2, -1, 2, -4, 8, ...

 Definition of a Geometric Sequence A Geometric Sequence is a sequence which the ratio of the common terms is equal. The general term is                     an  =  a1rn-1  where r is the common ratio.

Example

Find the general term of the geometric series such that

a5  =  48

and

a7  =  192

Solution

We have that

an  =  a1rn

which gives the equations

48  =  a1r4 ,         192  =  a1r6

Dividing the two equations, we get:

4  =  r2

Hence

r  =  2         or         r  =  -2

Substituting back into the first equation, we get

48  =  16a1

So that

a1 = 3

Hence the general term of the sequence is

an  =  (3)(2)n-1          or         an  =  (3)(-2)n-1

Geometric Series

 Theorem

Example

Find the sum

5 + 10 + 20 + 40 + ... + 2560

Solution:

a =  5     and     r  =  2     and     n  =  10

so that

a1(1 - rn)
Sn  =

(1 - r)

5(1 - 210)
=
1 - 2

=  5115

For an infinite geometric series if  |r| <1 then

Example

How much is going to taxes?  Suppose that we track a tax refund of \$100.  Each time money is spent 8% goes towards taxes and the rest gets spent again.  How much of the original \$100 will go back to taxes?

Solution

a1 = 8       r = 0.92    (The next amount to be taxed is 92% less than the current amount)

a1
S   =
1 - r

8               8
=                 =             =  \$100
1 - 0.92         0.08

Hence all of the refund will eventually find its way back to the government coffers.