Nicomachus’ Theorem

In 100 C.E., Nicomachus of Gerasa observed that

13 + 23 + 33 + … + n3 = (1 + 2 + 3 + … + n)2

Or “the sum of the cubes of 1 to n is the same as the square of their sum.” The diagram above demonstrates this neatly: Counting the individual squares shows that

1 × 12 + 2 × 22 + 3 × 32 + 4 × 42 + 5 × 52 + 6 × 62 = 13 + 23 + 33 + 43 + 53 + 63 = (1 + 2 + 3 + 4 + 5 + 6)2

http://www.futilitycloset.com/2014/12/30/nicomachus-theorem/