One of the greatest mathematicians in history, Karl Friedrich Gauss was asked a question by his teacher when he was 8 years old. The teacher told the class to sum up all the numbers from 1 to 100, like this: 1+2+3+4+...+98+99+100. The teacher thought this would keep them occupied for a few hours at the very least...When Gauss shouted 5050 the teacher fell back in his chair! Here is how the child did it:
He wrote two sums like this, one going forwards, the other going backwards:
\[1 + 2 + 3 + .... + 97 + 98 + 99 + 100 = S\]
\[100 + 99 + 98 + 97 + .... + 3 + 2 + 1 = S\]
Next he added them up:
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\[101 + 101 + 101 + ... + 101 = 2S\]
and noticed that there are 100 101's!
\[10100 = 2S\]
\[S = 5050\]
Not bad for an 8 year old!