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Gauss' Formula


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/   
    
Associated Topics:
High School Sequences, Series

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