Adding 1 to 100 the way Gauss did

Question

What is 1 + 2 + 3 + ... + 100? Find it in your head in a few seconds, then give the general formula for 1 + 2 + ... + n, explain why it works with a picture, and use the same idea to add 3 + 7 + 11 + ... + 99.

Accepted Answer

5,050. The pairing. 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, ..., 50 + 51 = 101. There are 50 pairs, each summing to 101: 50 × 101 = 5,050. The formula. For 1 to n, the same pairing gives n/2 pairs of n + 1 when n is even, and the general result is n(n + 1)/2. For odd n the middle number pairs with itself (as half a pair), and the formula still holds: 1 to 99 is 99 × 100 / 2 = 4,950, and adding 100 gives 5,050 again. A picture. Write the sum twice, once forwards and once backwards, and add the columns: each column is n + 1, there are n columns, so twice the sum is n(n + 1). Or draw a stairc…

Adding 1 to 100 the way Gauss did

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