GMAT Practice Problem: Sum of Squares

Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:

Problem

The sum of the first n positive perfect squares, where n is a positive integer, is given by the formula n3/3 + cn2 + n/6, where c is a constant. What is the sum of the first 15 positive perfect squares?

(A) 1,010
(B) 1,164
(C) 1,240
(D) 1,316
(E) 1,476

Solution

The brute-force way to solve this problem is literally to add up the first 15 positive perfect squares, from 1 to 225, inclusive. This is not necessarily completely out of bounds, given that we only have to sum up 15 numbers, all of which we should know already, and several of which are small. However, we should look for a shortcut using the formula.

Unfortunately, there is an unknown constant in the formula, but by using a small test number, we can solve for this constant. You can certainly pick n = 1, since it is a positive integer:
12 = 13/3 + c12 + 1/6
1 = 1/3 + c + 1/6
1/2 = c

If you feel uncomfortable picking n = 1, you can pick n = 2 and come to the same result almost as quickly.

Now, we plug n = 15 into the formula and solve:

12 + 22 + … + 152 = 153/3 + 152/2 + 15/6
= 15×15×15/3 + 15×15/2 + 15/6
= 15×15×5 + 15×15/2 + 5/2
= 225×5 + 225/2 + 5/2
= 1,125 + 230/2
= 1,125 + 115
= 1,240

The correct answer is (C).

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