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/
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