If n is a positive integer, which of the following is equivalent to n!

Question

If n is a positive integer, which of the following is equivalent to n! + (n + 1)! + (n + 2)! ? A. (n!)^3 B. 3(n + 1)! C. n!(n + 1)^3 D. n!(n + 2)^2 E. n!(n + 3) 2023-08-08_01-17-12.png

gmatophobia · Accepted Answer

Algebraic Approach

n! + (n + 1)! + (n + 2)!

n! ( 1 + (n + 1) + (n + 2)(n+1))

n! ( n + 2 + n^2 + 3n + 2)

n! ( n^2 + 4n + 4)

n! (n+2)^2

Substitution Approach

n = 1

n! + (n + 1)! + (n + 2)!

1 + (2)! + (3)! = 1 + 2 + 6 = 9

Answer choice elimination

A.  (n!)^3  →  =  1^3  = 1 : Eliminate

B.  3(n + 1)!  →  =  3 * 2!  = 6 : Eliminate

C.  n!(n + 1)^3  →  =  1 * 8  = 8 : Eliminate

D.  n!(n + 2)^2  →  =  1 * 9  = 9 : Keep

E.  n!(n + 3)  →  =  1 * 4  = 4 : Eliminate

Option D

JeffTargetTestPrep · Answer

Since n is a positive integer, the terms of the sum in the question stem are factorials.

n! + (n + 1)! + (n + 2)! = n! + n!(n + 1) + n!(n + 1)(n + 2) =

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

n!(n^2 + 4n + 4) =

n!(n + 2)^2

Answer: D

camahesh2896 · Answer

Let n = 1 so n! + (n+1)! + (n+2)! = 1+2+6 = 9,
Now put the value of n = 1 in the options to get 9 i.e Option D

NG24umich · Answer

Could someones comment as to why this didn't work/best way to go about this? I took this same approach thinking any value of n would work and am curious how to know to use 1 as my value.

Bunuel · Answer

In that version there was a typo. Please check corrected question in the first post and solutions above.

xyz02611 · Answer

How do we get from  n!+(n+1)!+(n+2)!  to  n! + n!(n + 1) + n!(n + 1)(n + 2) ?

For me the substitution approach is a quick way to solve this. But just wondering in case there are  variations to this type of question .. for instance if "n is a positive integer" is omitted..will the subsitution approach still work?

GMATinsight · Answer

n! + (n + 1)! + (n + 2)! Taking n! common n! + (n + 1)! + (n + 2)! = n!* = n!* = n!* = (n!*(n+2)^2 Answer: Option D --- GMATinsight | www.GMATinsight.com GMAT FOCUS PREP Expert Youtube: www.Youtube.com/GMATinsight

GmatKnightTutor · Answer

If n is a positive integer, which of the following is equivalent to  n! + (n + 1)! + (n + 2)!  ?

A.  (n!)^3

B.  3(n + 1)!

C.  n!(n + 1)^3

D.  n!(n + 2)^2

E.  n!(n + 3)

Hello, people. Let's get into this! A friendly way to do this question would be to plug in a positive integer value. If we let n = 1 in the original expression, we get:

1! + (1 + 1)! + (1 + 2)!
1! + 2! + 3!
1 + 2 + 6
 9

Which of the answer choices would, if we replace n with 1, equal 9?

(A) Would equal 1
(B) Would equal 6
(C) Would equal 8 
(D) Would equal 9
(E) Would equal 4

(D) is your answer.

Note: In this question, because only one of the answer choices gives the same value as the original expression, we can stop here. If, however, multiple answer choices did so, you would need to repeat the process with a different positive integer each time until you are left with only one option that always gives the same value as the original expression.

If n is a positive integer, which of the following is equivalent to n!

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If n is a positive integer, which of the following is equivalent to n!

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards