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A Summation Formula

Here is a formula for the sum of the first n natural numbers:


One example of this is tex2html_wrap_inline1171 . Gauss as a small child is said to have calculated this by noticing that 1+100=101, 2+99=101, 3+98=101, etc., so the entire sum is 50(101)=5050. We can use this idea to prove the formula. Let S be the sum. Then


Adding these together yields


So S=n(n+1)/2.

Carl Lee
Wed Apr 21 08:17:28 EDT 1999