These animations demonstrate the formulas for adding integers and their powers.

The advantage over other approaches is that they clearly illustrate the pattern which works for all powers,
and at the same time they illustrate the fundamental formulas for the integral and derivative of powers:

That rate of change of x^2 is 2x, and the rate of change of x^3 is 3x^2 and the
corresponding integral formulas is apparent in the addition
of two sides of length x or three sides of area x^2 to make successively larger squares and cubes.

The general formulas which are illustrated are

(n+1)^2=1+ 2 \sum_{j=1}^n j + 1 \sum_{j=1}^n 1

(n+1)^3=1+ 3 \sum_{j=1}^n j^2 + 3 \sum_{j=1}^n j + 1 \sum_{j=1}^n 1
 

etc.

which easily lead to

\sum_{j=1}^n j = n (n+1) / 2

\sum_{j=1}^n j^2 = n(n+1)(2n-1)/6 (?)

etc.