Formula for the Nth Term in a Geometric Sequence

Date: 08/05/99 at 00:29:13
From: Andy
Subject: Geometric Sequences

This is a complicated problem. Can you help me?

Tell whether each sequence is arithmetic or geometric, then find a 
formula for the nth term.

       2    4    6
      a    a    a
     ---, ---, ---, ...
      2    4    8

Please help me step by step in solving this equation.
Thank you.

Date: 08/05/99 at 11:59:07
From: Doctor Rob
Subject: Re: Geometric Sequences

Thanks for writing to Ask Dr. Math.

A sequence is geometric if the ratio of every pair of adjacent terms 
is a constant. A sequence is arithmetic if the difference of every 
pair of adjacent terms is a constant.

For a geometric sequence, if A is the first term, and R is the ratio 
of any two adjacent terms, then the nth term is given by the formula

     A*R^(n-1)

For an arithmetic sequence, if A is the first term, and D is the 
difference of any two adjacent terms, then the nth term is given by 
the formula

    A + D*(n-1)

Step 1: Find the ratios of adjacent terms, and the differences of 
adjacent terms.

Step 2: Figure out whether the ratios are all the same (R), or whether 
the differences are all the same (D), or neither, or both. (Yes, any 
of these can happen.)

Step 3: Get the values of A and either R or D, as appropriate.

Step 4: Substitute them in the appropriate formula above.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 08/05/99 at 22:37:33
From: Drew Chan
Subject: Re: Geometric Sequences

You gave me the steps to do this problem, but I still don't understand 
it. Can you please show me how to get the answer?

Thank you.

Date: 08/10/99 at 13:23:52
From: Doctor Rob
Subject: Re: Geometric Sequences

Step 1: Find the ratios of adjacent terms, and the differences of 
adjacent terms.

Ratios:
     (a^4/4)/(a^2/2) = a^2/2
     (a^6/8)/(a^4/4) = a^2/2

Differences:
     a^4/4 - a^2/2 = a^2/4*(a^2-2)
     a^6/8 - a^4/4 = a^4/8*(a^2-2)

Step 2: Figure out whether the ratios are all the same (R), or whether 
the differences are all the same (D), or neither, or both. (Yes, any 
of these can happen.)

Ratios are constant for any nonzero value of a, so R = a^2/2. 
Differences are constant only if a^2 = 2, and then D = 0. I assume 
that you are not interested in particular values of a, but only the 
general case, so the differences are not constant, and the sequence is 
not an Arithmetic Progression.

Step 3: Get the values of A and either R or D, as appropriate.

     A = a^2/2, R = a^2/2

Step 4: Substitute them in the appropriate formula above.

     A*R^(n-1)

is the right formula to use. You do the substitution and simplify to 
get the right answer.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/