Problem Solving

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j_shreyans wrote: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)1010
B)1164
C)1240
D)1316
E)1476

OAC

Our first task is to determine the value of c.
So, let's use the fact that, when n = 1, the SUM of the first 1 perfect squares is 1.

So, when n = 1, n³/3 + cn² + n/6 = 1
Replace n with 1 to get: 1³/3 + c(1²) + 1/6 = 1
Simplify: 1/3 + c + 1/6 = 1
Solve: c = 1/2

Great. The sum of the first n perfect squares = n³/3 + (1/2)n² + n/6 = 1

To find the sum of the first 15 perfect squares, plug n = 15 into the formula.
We get: 15³/3 + (1/2)15² + 15/6 = 1125 + 225/2 + 5/2
= 1125 + 230/2
= 1125 + 115
= 1240
= C

Cheers,
Brent

Hi j_shreyans,

Certain Quant questions on the GMAT require a bit of "playing around"; as Brent pointed out, you can figure out the value of C (the unknown constant in the question) by TESTing Values - TEST any number of perfect squares, sum them up and you can solve for C. With that value in place, you can then follow the rest of the prompt, plug in N=15 and solve.

It's important to remember that most questions CAN be solved in more than one way, so keep an open mind as to what else you might do to answer a given question. Here, we actually have a question that can also be solved without the given formula. The process would take a bit longer, but there are a couple of pattern-matching shortcuts that would save some serious time.

First, name the first 15 positive perfect squares (this is knowledge that you should have memorized before Test Day):
1
4
9
16
25

36
49
64
81
100

121
144
169
196
225

Don't add them up just yet. Notice that the "unit's digit" of the answers choices can only be one of 3 things (0, 4 or 6). Now, let's pair off those 15 numbers....

1 + 9 = ends in a 0
4 + 16 = ends in a 0
Notice a pattern.....
Every group of 5 values has a "1" and a "9" and a "4" and a "6", which means that we have lots of numbers that sum to a unit's digit of 0.

All that's left are the multiples of 5: 25, 100 and 225....those numbers, when summed, end in a 0.

This means that the total sum must also end in a 0. Eliminate B, D and E.

Now, let's estimate, starting with the larger values and working to the smaller ones....
225 + 196 = over 400
169 + 144 = over 300
121 + 100 + 81 = over 300
Total so far = over 1,000
And we have all those other values to add in....The total MUST be greater than 1010. Eliminate A.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich