If n is a positive integer, which of the following could be equal to

Question

If n is a positive integer, which of the following could be equal to  ((n+1)^3) - ((n^3)+1) ?

A.  7^3(6^3-1)

B.  7^3(6^3+1)

C.  2*26*27

D.  3*31*32

E.  0

GMATCoachBen · Accepted Answer

Video solution here:

https://www.youtube.com/watch?v=6UfutxEx2CI

Someone asked how common this is on the GMAT — expanding cubes like this is very rare. It saves a bit of time here, but is a low priority to memorize.

(n+1)^3 = n^3 + 3n^2 + 3n + 1

(Side note: the coefficients 1-3-3-1 here are the same as from "Pascal's Triangle" — I originally learned it because it's useful for the probability of how cards will split in games like bridge or hearts)

GMATGuruNY · Answer

An alternate approach is to test easy cases and look for a pattern:

If n=2, then  (n+1)^3 - (n^3+1) = 3^3 - (2^3 + 1) = 27 - 9 = 18 = 3*2*3 
If n=3, then  (n+1)^3 - (n^3+1) = 4^3 - (3^3 + 1) = 64 - 28 = 36 = 3*3*4 
If n=4, then  (n+1)^3 - (n^3+1) = 5^3 - (4^3 + 1) = 125 - 65 = 60 = 3*4*5

In every case, the result = (3)(product of two consecutive integers)
Only one answer choice matches this pattern.

D

gmatophobia · Answer

(n+1)^3 = n^3 + 1^3 + 3*n^2*1 + 3*1^2*n = n^3 + 1^3 + 3n^2 + 3n

Substituting the expanded form in the equation

n^3 + 1^3 + 3n^2 + 3n - (n^3 + 1)

n^3 + 1 + 3n^2 + 3n - n^3 - 1

3n^2 + 3n

3n(n+1)

Option D   matches with the above format for  n = 31

Kinshook · Answer

Asked: If n is a positive integer, which of the following could be equal to  ((n+1)^3) - ((n^3)+1) ?

((n+1)^3) - ((n^3)+1) = (n+1)ˆ3 - nˆ3 - 1 = 3nˆ2 + 3n = 3n(n+1)

IMO D

GMATinsight · Answer

ALGEBRIC PROPERTIES REQUIRED:   
  Property 1:    (a^3+b^3) = (a+b)*(a^+b^2-ab) 
  Property 2:    (a+b)^2 = (a^2+b^2+2ab)

((n+1)^3) - ((n^3)+1) = ((n+1)^3) - ((n^3)+1^3) = ((n+1)^3) - (n+1)*(n^2+1^2-n*1)

=(n+1)  =(n+1)  = (n+1)  = 3*n*(n+1)

i.e. 3*product of two consecutive numbers

Answer: Option D

MartyMurray · Answer

If n is a positive integer, which of the following could be equal to  ((n+1)^3) - ((n^3)+1) ?

The fact that this question involves  (n+1)^3  can make it seem intimidating, but as is often the case with GMAT Quant questions, if we just try something logical, we can get to the answer.

Expanding  (n+1)^3  turns out not to be a big deal:

(n + 1)^2 = n^2 + 2n + 1

(n + 1)^3 = (n^2 + 2n + 1 )(n + 1) = n^3 + 3n^2 + 3n + 1

So,  ((n+1)^3) - ((n^3)+1) = ( n^3 + 3n^2 + 3n + 1) - ((n^3)+1) = 3n^2 + 3n

Since  n  is a positive integer, we can eliminate (E).

Then, since  3n^2 + 3n  is a multiple of  3 , we can eliminate (A) and (B) since ( 7  raised to a power) × ( 1  less or more than a multiple of  6 ) cannot be a multiple of  3 .

Then, (D) signals that it could make sense to factor the  3  out of  3n^2 + 3n . Doing so, we get  (3)(n^2 + n) , which we can quickly recognize as  (3)(n)(n + 1) .

That perfectly matches (D), which is  (3)(31)(31 + 1) .

A.  7^3(6^3-1)

B.  7^3(6^3+1)

C.  2*26*27

D.  3*31*32

E.  0

Correct answer:   D

KarishmaB · Answer

Using basic algebraic identities,
 (a+b)^3 = a^3 + b^3 + 3ab(a+b) , we get 
 ((n+1)^3) - ((n^3)+1) = n^3 + 1 + 3n(n+1) -n^3 - 1 
 = 3n(n+1) 
Hence whatever we get must be of this form.

Options (A) and (B) do not have a 3 in them. 
Option (C) is 2n(n+1). 
Option (D) works.
Option (E) is not possible because n is a positive integer.

Answer (D)

If n is a positive integer, which of the following could be equal to

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If n is a positive integer, which of the following could be equal to

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