Sums of the first n numbers, squares

Problem: Find the sum of the first 1000 natural numbers.

Solution: Write the numbers twice as follows:

 \begin{matrix} 1 & + & 2 & + & \dots & + & 999 & + & 1000 \\ 1000 & + & 999 & + & \dots & + & 2 & + & 1 \end{matrix}

Summing the numbers in each column, we have:

2 (1 + 2 + \dots + 1000) = 1001 + 1001 + \dots + 1001 (1000 times)

And dividing by two we have

1 + 2 + \dots + 1000 = \frac{1000 * 1001}{2} = 500500

This method clearly works not just for 1000, but any natural number n.

Problem: Find the sum of the squares of the first n natural numbers

Solution: Arrange the numbers three times, in triangles as follows:

Arranging the numbers in triangles, where each summand is successively rotated 120 degrees.

Here we prove what is obvious from the picture: that the numbers in the “sum” triangle work out as expected.

In the leftmost triangle, there are i copies of i in the ith row (indexing from 1), so the sum of the ith row is i^2, and the sum of all the numbers in the triangle is (1^2 + 2^2 + \dots + n^2). Since the other two “summand” triangles are just rotations of the first triangle, their sums are also the sum of the first n squares.

In the “sum” triangle at the ith row j spaces from the left (indexing both from zero), we have the value (1+i) + (n-i+j) + (n-j) = 2n+1. In each triangle there are 1 + 2 + \dots + n = \frac{n(n+1)}{2} different numbers (as per the first proof), so there are that many copies of 2n+1 in the fourth triangle. Summing up the fourth triangle gives three times our desired result, and hence

3(1^2 + 2^2 + \dots + n^2) = \frac{n(n+1)(2n+1)}{2}, or
(1^2+2^2+\dots + n^2) = \frac{n(n+1)(2n+1)}{6}

This second proof was shown to me by Sándor Dobos, a professor of mine in Budapest whom I remember fondly.

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4 thoughts on “Sums of the first n numbers, squares

  1. Pingback: Methods of Proof — Induction | Math ∩ Programming

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  3. Pingback: Deconstructing the Common Core Mathematical Standard | Math ∩ Programming

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