Discussion in practice: Gauss sum

When Gauss, who later became a famous mathematician, was still in school, his teacher asked him to sum the integers from 1 to 100. The teacher probably wanted to punish him with a boring task. However, after a few seconds he answered: 5050.

In the next exercise we see how Gauss got this result. Can you think of any alternative ways to reach the conclusion?

Exercise 3.

For (ninmathbb N,, nge 1,) let (P(n)) be the proposition [P(n):quad 1+2+cdots + n=dfrac{n(n+1)}2.]

1) Verify that (P(1), P(2)) and (P(3)) are true.

2) Prove the result by mean of Gauss method as follows:

i) Call (S(n)=1+2+cdots +n). Write in two subsequent rows the sum (S(n)) by reversing the order of its terms: [(a)quad S(n)=1+2+cdots+n] [(b)quad S(n)=n+(n-1)+cdots +1] Notice that (1+n=2+(n-1)=3+(n-2)=…).

ii) Sum term by term on the columns in (a) and (b): you get (2S(n)=cdots)