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

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nick13
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If n is a positive integer, which of the following could be equal to [#permalink] New post   09 Oct 2023, 21:32
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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. \(0\)
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D
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Re: If n is a positive integer, which of the following could be equal to [#permalink] New post   09 Oct 2023, 22:15
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nick13 wrote:
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


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

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 \(n = 31\)
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Re: If n is a positive integer, which of the following could be equal to [#permalink] New post   10 Oct 2023, 00:25
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
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If n is a positive integer, which of the following could be equal to [#permalink] New post   17 Dec 2023, 18:27
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Video solution here:



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)
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If n is a positive integer, which of the following could be equal to [#permalink]
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