Consider the numerical pyramid below, formed by simply putting down the series of odd numbers into a pyramid.
7 9 11
13 15 17 19
. . .
Find a formula for the sum of the numbers in the nth row, and prove it.
The first number of the n-th row is twice the (n-1)th triangular number, plus one. The last number of the same row is twice the n-th triangular number, minus one. There are n numbers in the n-th row.
Thus, the sum of the numbers of the n-th row works out to be (n-1)n+1 plus n(n+1)-1, times n, which is n cubed.