Sum of the First n Natural Numbers
Sum of First n Natural Numbers We prove the formula 1+ 2+ ... + n = n(n+1) / 2 , for n a natural number.

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.

This text was written by Joseph Goguen in May 1996.
To the first tatami (i.e., proof) page.
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This page was generated by Kumo on Fri Jul 02 14:09:10 PDT 1999.