Sum the integers from 1 to 1000

Question

Solve in your head. What is 1 + 2 + 3 + \dots + 1000? Aim: under 5 seconds. Follow-up: what is 1 + 2 + \dots + 999?

Accepted Answer

Use 1 + 2 + \dots + n = \frac{n(n+1)}{2} with n = 1000: \frac{1000 	imes 1001}{2} = \frac{1{,}001{,}000}{2} = 500500. The pairing picture: 1 + 1000 = 1001, and there are 500 such pairs, so 500 	imes 1001 = 500500. Follow-up. The sum to 999 is just the sum to 1000 minus the last term: 500500 - 1000 = 499500. (Or directly, \frac{999 	imes 1000}{2} = 499500.) Takeaway. The formula \frac{n(n+1)}{2} scales to any size; you never add terms one by one. Dropping or adding an endpoint just subtracts or adds that single value.

Sum the integers from 1 to 1000

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