Simplify a Geometric Series

Date: 05/06/2003 at 16:51:02
From: sharath
Subject: Formula needed

Can you please tell me the formula for

   x^n + x^(n - 1) + x^(n - 2) + ... + x^(n - n)

I don't know how to condense this formula. I came to it while trying 
to find out the 'inflation' effect on each year's savings.

Thanks.

Date: 05/07/2003 at 02:54:25
From: Doctor Luis
Subject: Re: Formula needed

Hi Sharath.

If you look at your equation carefully, you'll notice that n-n = 0. In 
fact, the exponent decreases as you keep adding more terms. You can 
rewrite them like this:

 x^0 + x^1 + x^2 + ... + x^(n - 2) + x^(n - 1) + x^n

Of course, x^0 = 1, and x^1 = x

This sum is generally known as a geometric series. Finding a simpler 
form of this formula is easy, although a little tricky.

First, you name the sum:

   S = 1 + x + x^2 + ... + x^(n-2) + x^(n-1) + x^n

Then you multiply by x,

  xS = x + x^2 + ... + x^(n-1) + x^n + x^(n+1)

That looks kind of similar to S. In fact, you can see the similarity
more closely if you write it like this:

  xS = -1 + 1 + x + x^2 + ... + x^(n-1) + x^n + x^(n+1)
     = -1 + (1 + x + x^2 + ... + x^(n-1) + x^n) + x^(n+1)

The terms inside the parenthesis are just S. Therefore,

 xS  = -1 + (1 + x + x^2 + ... + x^(n-1) + x^n) + x^(n+1)

 xS  = -1 + S + x^(n+1)

This last equation can be solved for S, which will give a much simpler
condensed form for the geometric series.

I hope this helped! Let us know if you have any more questions.

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