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 

        a_n  =  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

        a_n  =  (3)(2)^n-1          or         a_n  =  (3)(-2)^n-1  

Geometric Series

 

Theorem

Example

Find the sum

        5 + 10 + 20 + 40 + ... + 2560

Solution:     

        a₁  =  5     and     r  =  2     and     n  =  10 

so that

                     a₁(1 - rⁿ)
        S_n  =                     
                        (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.

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