How to Add 1 to N in One Step: the Gauss Sum

Sum 1 through N in one step: N × (N+1) ÷ 2.

Boss Robo

1 + 2 + 3 + ... + 100 = 5050

  1. Gauss, aged nine, folded the list end to end: 1 + 100, 2 + 99, all 101.

    1100101299101398101………5051101+

  2. A hundred numbers fold into fifty pairs, so the total is 50 × 101 = 5050.

    100 numbers pairs
    50 × 101 = 5050

  3. Any run folds the same way: N / 2 pairs of N + 1, so the sum is N(N + 1) / 2.

    5 + 1 = 655 × 6 ÷ 2 = 15

How do you add up 1 to 100 in one step?

To add every whole number from 1 to N, multiply N by N + 1 and halve the result. For 1 to 100, that is 100 × 101 ÷ 2 = 5050.

It works because the list folds in half: 1 pairs with 100, 2 with 99, 3 with 98, and every pair adds to 101.

There are 100 numbers, so there are 50 pairs, and 50 × 101 = 5050. Writing the list twice, once backwards, also handles an odd count: each column adds to N + 1, there are N columns, and the two copies make twice the sum. So for 1 to 15, the sum is 15 × 16 ÷ 2 = 120, even though 15 numbers do not split into pairs. The same idea handles a list that does not start at 1: 20 + 21 + … + 30 has 11 numbers, and the first and last add to 50, so the sum is 11 × 50 ÷ 2 = 275.

Coach Pi

Your turn

Three to try in your head. Tap what you get.

1 + 2 + ... + 75 = ?

1 + 2 + ... + 30 = ?

1 + 2 + ... + 10 = ?

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