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Sum of the 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.

To the Kumo demos homepage.
To the Tatami Project homepage.
To the UCSD Meaning and Computation Group homepage.

       This page was generated by Kumo on Tue Feb 13 17:31:26 PST 2001.