Sum the integers from 50 to 100

Question

Solve in your head. What is 50 + 51 + 52 + \dots + 100? Aim: under 10 seconds.

Accepted Answer

Same idea, count times average, but the trap is the term count. From 50 to 100 inclusive there are 100 - 50 + 1 = 51 terms (the "+1" catches both endpoints). The average term is \frac{50 + 100}{2} = 75, so 50 + 51 + \dots + 100 = 51 	imes 75 = 3825. Cross-check by subtraction: \big(1 + \dots + 100\big) - \big(1 + \dots + 49\big) = 5050 - 1225 = 3825. Both agree. Takeaway. For a range from a to b, the count is b - a + 1, not b - a, and the sum is 	ext{count} 	imes \frac{a+b}{2}. When in doubt, verify with "full sum minus the head you chopped off."

Sum the integers from 50 to 100

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