Date: Apr 1, 2013 12:42 PM
Author: dan.ms.chaos@gmail.com
Subject: Re: 1 + 2 + ... + n a polynomial how?

On Apr 1, 7:18 pm, Jussi Piitulainen <jpiit...@ling.helsinki.fi>
wrote:
> Is it obvious that 1 + 2 + ... + n is a polynomial of degree 2? How?
>
> I mean the sum of the first n positive integers. I would like to see
> that it is a polynomial of degree 2 _without using_ the fact that it
> is equal to n(n + 1)/2. Zeilberger (his new Opinion 129) says Gauss
> could have used the polynomiality of the sum to support the equality,
> rather than the other way around.
>
> Thanks for any insight.


The difference between two consecutive terms of the sum series , S(n)
- S(n-1) = n , a first degree polynomial . Ergo, the sum itself is a
second degree polynomial .