Simplifying [ (n! / (n-1) !] gives: O (n+1) (n+2)! n!
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Simplifying [ (n! / (n-1)!) ] gives:
- ⃝ (n+1)
- ⃝ (n+2)!
- ⃝ n
- ⃝ n!
**Explanation:**
To simplify [ (n! / (n-1)!) ], we need to understand the factorial function. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Mathematically, it is defined as:
\[
n! = n \times (n-1) \times (n-2) \times \ldots \times 1
\]
Now, let's simplify the term \( \frac{n!}{(n-1)!} \):
Since \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \) and \( (n-1)! = (n-1) \times (n-2) \times \ldots \times 1 \), we can rewrite \( n! \) as \( n \times (n-1)! \).
Thus,
\[
\frac{n!}{(n-1)!} = \frac{n \times (n-1)!}{(n-1)!}
\]
The \( (n-1)! \) terms cancel out:
\[
\frac{n \times (n-1)!}{(n-1)!} = n
\]
Therefore, the correct answer is:
\[
n
\]
So, the correct option from the given multiple choices is:
- ⃝ n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd529b6da-11b0-40e5-85fb-f211e59fb34e%2F2021a6d4-9da4-4573-b29e-7f3e328fac8f%2Fpwbagp.png&w=3840&q=75)
Transcribed Image Text:### Simplifying [ (n! / (n-1)!) ] gives:
- ⃝ (n+1)
- ⃝ (n+2)!
- ⃝ n
- ⃝ n!
**Explanation:**
To simplify [ (n! / (n-1)!) ], we need to understand the factorial function. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Mathematically, it is defined as:
\[
n! = n \times (n-1) \times (n-2) \times \ldots \times 1
\]
Now, let's simplify the term \( \frac{n!}{(n-1)!} \):
Since \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \) and \( (n-1)! = (n-1) \times (n-2) \times \ldots \times 1 \), we can rewrite \( n! \) as \( n \times (n-1)! \).
Thus,
\[
\frac{n!}{(n-1)!} = \frac{n \times (n-1)!}{(n-1)!}
\]
The \( (n-1)! \) terms cancel out:
\[
\frac{n \times (n-1)!}{(n-1)!} = n
\]
Therefore, the correct answer is:
\[
n
\]
So, the correct option from the given multiple choices is:
- ⃝ n
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