In elementary school, Carl Gauss (1777 - 1855) was asked to sum the numbers from 1 to 100. The teacher was probably expecting a few
minutes of quiet, but Gauss produced the answer in seconds.
He probably divided the set of 100 numbers into 50 pairs of 101:
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101, etc.
Hence, the sum of 1 + 2 + 3 + 4 + ... + 99 + 100 = 50 (101) = 5050.
Algebra students can now derive the formula for the sum of the first n integers
by using Gauss' trick:
Let S = 1 + 2 + 3 + 4 + ... + (n-2) + (n-1) + n
Then S = n + (n-1) + (n-2) + (n-3) + ... + 3 + 2 + 1
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2S = (n+1) + (n+1) + (n+1) + (n+1) + ... + (n+1) + (n+1) + (n+1)
2S = n (n + 1)
S = n (n + 1) / 2
Now, back to Gauss, and the rest of the story ...
School master: Class, I want each of you to spend the next 15 minutes summing the
integers
from 1 to 100. This is difficult and I'm not sure of what
your chances are.
Gauss: Fifty-fifty?
School master: Incredible! You calculated the correct sum immediately!
You're a genius!