Date: 02/03/99 at 08:25:09 From: Roxanne Stern Subject: Gauss's Formula My class has been using Gauss' formula recently. We were able to find the number of dots in a triangle with 1, 2, 3, and 4 layers (i.e.: 1, 3, 6, 10, ...) by using Gauss' formula. However, when we put the triangles together in a stack 10 high of oranges, with the last stack containing 1 orange, we faced a problem. By doing this, we saw that we want to add up the numbers we found earlier (1 + 3 + 6 + ... .) Some of the kids tried to use Gauss' formula and came up with 280 (from 10(1+55)/2). The ones who added 1 + 3 + 6 + 10 + ... + 55 by hand came up with 220 (the right answer). Why did Gauss' formula not work this time? Does Gauss' formula work only for any progression? What did we do wrong? Thanks for your help.
Date: 02/03/99 at 09:59:22 From: Doctor Rob Subject: Re: Gauss's Formula What is wrong here is that Gauss's formula works only for arithmetic progressions, that is, ones for which each term is gotten from its predecessor by adding the same constant d. The sequence 1, 2, 3, 4, ... is an arithmetic progression with d = 1. The sequence 1, 3, 6, 10, ... is not an arithmetic progression at all. That means that Gauss's formula does not apply, so using it will give nonsensical answers. Gauss's formula is equivalent to this: a + (a+d) + (a+2*d) + .. + (a+[k-1]*d) = k*(a + a+[k-1]*d)/2 = a*k + d*k*(k-1)/2 To add up the partial sums of this sum, use this formula: a + (2*a+d) + (3*a+3*d) + (4*a+6*d) + ... + (a*k+d*k*[k-1]/2) = a*k*(k+1)/2 + d*k*(k-1)*(k+1)/6 = (3*a+[k-1]*d)*k*(k+1)/6 In your case, a = d = 1 and k = 10, so the sum is (3+9)*10*11/6 = 220. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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