What is the sum of the squares of the integers from 11 to 30 inclusive? Give a second method that checks the first.

What is the sum of the squares of the integers from 11 to 30 inclusive? Give a second method that checks the first.

Approach: Apply the closed form for the sum of the first n squares at both endpoints and subtract, then verify with the relation between mean square, squared mean and variance.

9070. Use the formula n(n + 1)(2n + 1)/6 and take the difference of two partial sums. For n = 30 it gives 30 * 31 * 61/6 = 5 * 31 * 61 = 9455, dividing the 6 into the 30 before multiplying anything. For n = 10 it gives 10 * 11 * 21/6 = 385. The difference is 9455 - 385, which is 9070. Cancelling the 6 into the largest convenient factor first keeps every intermediate number small, which is what makes the formula usable under a clock. The variance identity checks it exactly: the sum of 20 squares is 20 times the mean square, and the mean square is the squared mean plus the variance. The variance of 20 consecutive integers is 33.25, so 20 * (420.25 + 33.25) = 20 * 453.5 = 9070.

Follow-up: What is the sum of the cubes from 11 to 30, and what identity links it to the sum of the integers?

Key concepts: sum of squares formula, difference of partial sums, variance identity.