Skip main navigation

Discussion in practice: Gauss sum

Gauss sums

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)

This article is from the free online

Precalculus: the Mathematics of Numbers, Functions and Equations

Created by
FutureLearn - Learning For Life

Reach your personal and professional goals

Unlock access to hundreds of expert online courses and degrees from top universities and educators to gain accredited qualifications and professional CV-building certificates.

Join over 18 million learners to launch, switch or build upon your career, all at your own pace, across a wide range of topic areas.

Start Learning now