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Worked solution

Explain the pairing behind the sum from one to one hundred

In the source story, Gauss rapidly finds 1+2+3+⋯+98+99+100=50501+2+3+\cdots+98+99+100=5050. How does pairing terms explain the result? What pattern makes the calculation quick?

CJC · Sequences and Series · Unnumbered Gaussian-method question, page 29

Answer

Add the forward and reversed sums: 2S=100(101)2S=100(101), so S=5050S=5050.

Compare your reasoning

Write the sum in both orders

S=1+2+3+⋯+98+99+100(1),S=100+99+98+⋯+3+2+1(2).\begin{aligned}S&=1+2+3+\cdots+98+99+100&&\text{(1)},\\S&=100+99+98+\cdots+3+2+1&&\text{(2)}.\end{aligned}

Reversing a finite sum does not change its value. These are the source equations (1) and (2).

Pair terms in corresponding positions

2S=(1+100)+(2+99)+(3+98)+⋯+(98+3)+(99+2)+(100+1)=100(101)\begin{aligned}&2S\\ & =(1+100)+(2+99)\\ &{}+(3+98)+\cdots+(98+3)\\ &{}+(99+2)+(100+1)\\ & =100(101)\end{aligned}

There are one hundred brackets, each equal to 101. This adds two copies of the original sum.

Divide by two

2(1+2+3+⋯+98+99+100)=10100,1+2+3+⋯+98+99+100=50502(1+2+3+\cdots+98+99+100)=10100,\qquad1+2+3+\cdots+98+99+100=5050

The source emphasises pattern recognition: equal pair sums replace a long calculation.

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