## Sum of powers of positive integers

Sums of the form:
\[
\sigma_k(n)=1^k+2^k+\cdots+n^k
\]
are called the sum of powers of postive integers and have been studied throughout the history of mathematics. In this section we will give a historical account of these studies.

### Pythagoras (570-500 BC)

Pythagoras observed that the sum of succesive powers of positive integers could be arranged as triangles. They called these sum triangular numbers.

Triangular Numbers |

He noticed that two times the same triangular gives the number $n(n+1)$.

So concluded:
\[
\sigma_1(n)=1+2+3+\cdots+n=\frac{n(n+1)}{2}
\]
He also noticed that the sum of two consecutive triangular numbers is a square number.

This can also be algebraically verified:
\[
\frac{n(n+1)}{2}+\frac{(n+1)(n+2)}{2}=(n+1)^2
\]

In modern analysis courses these formulas are proofed using the induction method.

### Archimedes (287 - 212 BC)

Archimedes studied the sum of powers of squares. He noticed following:

\[
3(1^2+2^2+3^2+\cdots+n^2)=n^2(n+1)+(1+2+3+\cdots+n)
\]

Consider the sum of the following serie:
\[
s(n)=1+3+6+10+\cdots+\frac{n(n+1)}{2}
\]
Notice that geometrically this sum can be represented by a pyramid of $s(n)$ cubes.

If we add 6 of these pyramides together we have a block with $n(n+1)(n+2)$ cubes.

So we have found:
\[
s(n)=1+3+6+10+\cdots+\frac{n(n+1)}{2}=\frac{n(n+1)(n+2)}{6}
\]

## References

**[1]** Janet Beery, The sum Of Powers of Positive Integers, MathDL.

Copyright ©2012 Jacq Krol. All rights reserved. Created June 2012; last updated August 2012.