ctrlaltdel wrote:
The sum of the squares of the first 15 positive integers (1^2 + 2^2 + 3^2 + . . . + 15^2) is equal to 1240. What is the sum of the squares of the second 15 positive integers (16^2 + 17^2 + 18^2 + . . . + 30^2) ?
(A) 2480
(B) 3490
(C) 6785
(D) 8215
(E) 9255
How do we solve this one?
If you know the sum of squares formula (something I've never seen tested in a real GMAT question), you can use it to sum the first 30 squares, then subtract 1240 to get the answer. The sum of the squares of the first n positive integers is given by (n)(n+1)(2n+1)/6, though I'd be very surprised to find a real GMAT question where this was at all helpful to know.
Or, if you're familiar with remainder/modular arithmetic (if not, this solution might seem confusing), you might notice that when you add any three consecutive perfect squares, the remainder will be 2 when you divide by 3 (since one of the squares will be divisible by 3, and the other two will both give a remainder of 1). So here we're adding five sets of three numbers, where each set gives a remainder of 2 when divided by 3. Since we can add remainders, the remainder when we divide 16^2 + 17^2 + ... + 30^2 by 3 will be the remainder when we divide 5*2 = 10 by 3, so the remainder will be 1. Of the answer choices, only B and D give the correct remainder, but B is far too small.
You could also estimate here. Notice that 15^2 + 16^2 + 17^2 + 18^2 + 19^2 is quite a bit bigger than 15^2 + 15^2 + ... + 15^2 = 5*15^2, and quite a bit smaller than 20^2 + 20^2 + ... + 20^2. Similarly, the next block of five squares add to something quite a bit bigger than 5*20^2, and quite a bit smaller than 5*25^2, and similarly for the last block. So our sum S must satisfy the following inequality, and shouldn't be anywhere close to the extreme low or high values:
5*15^2 + 5*20^2 + 5^25^2 < S < 5*20^2 + 5*25^2 + 5*30^2
5(15^2 + 20^2 + 25^2) < S < 5(20^2 + 25^2 + 30^2)
5(1250) < S < 5(1925)
6250 < S < 9625
and since answer choice C is too close to the low end of this range, and E too close to the high end, D must be correct. If one is willing to do more elaborate computations, one can easily get an estimate to narrow the above range further, of course.
Or you can proceed algebraically, using the (x +y)^2 = x^2 + 2xy + y^2 pattern to relate the sum to the one given in the question:
16^2 + 17^2 + 18^2 + ... + 29^2 + 30^2 =
(1 + 15)^2 + (2 + 15)^2 + (3 + 15)^2 + ... + (14 + 15)^2 + (15 + 15)^2 =
(1^2 + 2*15 + 15^2) + (2^2 + 2*2*15 + 15^2) + (3^2 + 2*3*15 + 15^2) + ... + (14^2 + 2*14*15 + 15^2) + (15^2 + 2*15*15 + 15^2) =
(1^2 + 2^2 + ... + 15^2) + 2*15(1 + 2 + 3 + ... + 14 + 15) + 15*15^2
= 1240 + 3600 + 3375
= 8215