BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

n positive perfect square

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Legendary Member
Posts: 510
Joined: Thu Aug 07, 2014 2:24 am
Thanked: 3 times
Followed by:5 members

n positive perfect square

by j_shreyans » Mon Sep 15, 2014 4:04 am
The sum of the first n positive perfect squares, where n is a positive integer, is given by the formula n^3/3+cn^2+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
Top
Source: — Problem Solving |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Mon Sep 15, 2014 6:57 am
Adding a few SPACES and some NICER NOTATION will make this question less ambiguous.
For more on posting ambiguous-free expressions see: https://www.beatthegmat.com/read-this-fi ... 79221.html

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
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800

by [email protected] » Tue Sep 16, 2014 11:14 am
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
Contact Rich at [email protected]
Image
Top
Post new topic Post Reply
• Page 1 of 1