Sum the integers from 1 to 200

Question

Solve in your head. What is 1 + 2 + 3 + \dots + 200? Aim: under 5 seconds. Follow-up: what is 1 + 3 + 5 + \dots + 199?

Accepted Answer

Pair the ends: 1 + 200 = 201, 2 + 199 = 201, and so on. There are 100 such pairs, so 1 + 2 + \dots + 200 = 100 	imes 201 = 20100. Equivalently, (number of terms) 	imes (average term) = 200 	imes 100.5 = 20100. Follow-up. The odd numbers 1, 3, 5, \dots, 199 are the first 100 odd numbers, and the sum of the first n odd numbers is exactly n^2. So the answer is 100^2 = 10000. (Check: 50 pairs like 1 + 199 = 200, giving 50 	imes 200 = 10000.) Takeaway. Any arithmetic series is 	ext{count} 	imes 	ext{average of first and last}. Two shortcuts worth memorizing: 1 + \dots + n = \frac{n(n+1)}{2}…

Sum the integers from 1 to 200

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