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Toolkit 56

Gauss' Idea (Rainbow Idea)

Pair up the elements:

Gauss pairing 1+100, 2+99, ... equals 50 x 101

or

S=1+2+3+⋯+98+99+100S=1+2+3+\cdots+98+99+100
S=100+99+98+⋯+3+2+1S=100+99+98+\cdots+3+2+1
2S=101+101+⋯+1012S=101+101+\cdots+101
2S=100×1012S=100\times101

Therefore:

S=100×1012S=\frac{100\times101}{2}
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