There is a simple applet showing the essence of the inductive proof of this result. To run this applet, you first enter the number n you wish to have illustrated; space limitations require 0<n<11. Then push the [Next] button to step through the stages of the proof.
The base case shown by the applet is n=1, although on the proof pages the base case is n=0; this is just because there is nothing to show when n=0.
Assuming the result for n means we know how to sum half of an nx(n+1) rectangle having rows with 1, 2, ..., n red dots, respectively. Now we add a new row with all black dots, and then one more red dot to each row. The result is another figure of the same form, but with the parameter n+1 instead of n.