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### Gauss' Formula

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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

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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