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D
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Difficulty:
15%
(low)
Question Stats:
82% (01:44) correct
18%
(01:51)
wrong
based on 913
sessions
History
Date
Time
Result
Not Attempted Yet
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 [ 26.32 KiB | Viewed 24745 times ]
A. \((n!)^3\)
B. \(3(n + 1)!\)
C. \(n!(n + 1)^3\)
D. \(n!(n + 2)^2\)
E. \(n!(n + 3)\)
Attachment:
2023-08-08_01-17-12.png [ 26.32 KiB | Viewed 24745 times ]
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07 Aug 2023, 12:27
Kudos
Bookmarks
Bunuel
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
27 Aug 2023, 06:48
Kudos
Bookmarks
Bunuel
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
