The sum of the squares of the first 15 positive integers (1^2 + 2^2 + 3^2 + . . . + 15^2) is equal to 1270. What is the sum of the squares of the second 15 positive integers (16^2 + 17^2 + 18^2 + . . . + 30^2) ?	Choose the correct choice from the following options:
A) 2480	B) 3490	C) 6785	D) 8245	E) 9255
You'll never need a formula for the sums of squares on the actual GMAT. You don't need to use that formula here, though it's not all that straightforward to solve without one. Two different approaches:   16^2 + 17^2 + 18^2 + ... + 30^2 = (15 + 1)^2 + (15 + 2)^2 + (15 + 3)^2 + ... + (15 + 15)^2   Now we can expand each square; they are all in the (x + y)^2 = x^2 + 2xy + y^2 pattern.   = (15^2 + 2*15 + 1^2) + (15^2 + 4*15 + 2^2) + (15^2 + 6*15 + 3^2) + ... + (15^2 + 30*15 + 15^2)   Now we have fifteen 15^2 terms, so adding these gives 15*15^2 = 15^3 = 3375. We also have the entire sum 1^2 + 2^2 + ... + 15^2, which we know is equal to 1240. Finally adding the middle terms, we have:   2*15 + 4*15 + 6*15 + ... + 30*15 = 15(2 + 4 + 6 + .... + 30) = 15*2*(1 + 2 + 3 + ... + 15) = 15*2*8*15= 
<gadget id="calculator">15*2*8*15</gadget>
<output>3_600</output>
3600   So the sum must be  3375 + 1240 + 3600= 
<gadget id="calculator"> 3375 + 1240 + 3600</gadget>
<output>8_215</output>
8215   Alternatively, we can use a different factoring pattern. We want to find the value of 30^2 + 29^2 + ... + 17^2 + 16^2. Well if we subtract 15^2 + 14^2 + .... + 2^2 + 1^2 from this, the answer will be 1240 less than what we want to find. So if we can find the value of   30^2 + 29^2 + ... + 17^2 + 16^2 - (15^2 + 14^2 + .... + 2^2 + 1^2)   then we can add 1240 to get the answer. Now grouping the terms above to get differences of squares, we have= (30^2 - 15^2) + (29^2 - 14^2) + ... + (16^2 - 1^2)   and factoring each of these using x^2 - y^2 = (x + y)(x - y), we have   = 45*15 + 43*15 + 41*15 + ... + 17*15 = 15(45 + 43 + 41 + ... + 17)   In brackets we have an equally spaced sum with fifteen terms, which we can evaluate using the familiar formula. So the above equals     15*15*62/2= 
<gadget id="calculator">  15*15*62/2</gadget>
<output>6_975</output>
6975   and adding back the 1270, we get the answer of 8245. (ANS 8245)
<result>D</result>
The sum of the squares of the first 15 positive integers (1^2 + 2^2 + 3^2 + . . . + 15^2) is equal to 1270. What is the sum of the squares of the second 15 positive integers (16^2 + 17^2 + 18^2 + . . . + 30^2) ?