Friday, 6 October 2017

algebra precalculus - Sum of the first natural numbers: how many and what are the most common methods to verify it?

We know that Gauss has shown that the sum S of the first n natural numbers is given by the relation:



S=n(n+1)2


The proof that I remember most frequently is as follows:




Let be S=1+2++(n1)+n We can write S it also as: S=n+(n1)++2+1.
By adding up member to member we get:
2S=(n+1)+(n+1)++2+1ntimes.
Hence we obtain the ().



How many other simple methods exist to calculate the sum of the first natural numbers?

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