Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Simplify**
\[
\frac{(n-2)!}{(n+4)!}
\]
This expression involves factorials in the numerator and the denominator. To simplify, note that the factorial of a number \( n \), denoted by \( n! \), is the product of all positive integers from 1 to \( n \). Thus, \((n+4)!\) can be expanded as \((n+4) \times (n+3) \times (n+2) \times (n+1) \times n \times (n-1) \times \ldots \times 1\).
Therefore, \((n-2)!\) cancels with part of \((n+4)!\), leaving:
\[
\frac{1}{(n+4) \times (n+3) \times (n+2) \times (n+1) \times n \times (n-1) \times (n-2) \times \ldots \times 1}
\]
This results in:
\[
\frac{1}{(n+4) \times (n+3) \times (n+2) \times (n+1) \times n}
\]
For educational purposes, it's important to understand the concept of factorials and how they can be expanded and canceled in fractions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59fae379-c2a2-4553-824b-3c3ac08e75aa%2Fe2d9d8c2-cd2b-4f6f-a624-8b529557bcd6%2Fwh7tsgr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Simplify**
\[
\frac{(n-2)!}{(n+4)!}
\]
This expression involves factorials in the numerator and the denominator. To simplify, note that the factorial of a number \( n \), denoted by \( n! \), is the product of all positive integers from 1 to \( n \). Thus, \((n+4)!\) can be expanded as \((n+4) \times (n+3) \times (n+2) \times (n+1) \times n \times (n-1) \times \ldots \times 1\).
Therefore, \((n-2)!\) cancels with part of \((n+4)!\), leaving:
\[
\frac{1}{(n+4) \times (n+3) \times (n+2) \times (n+1) \times n \times (n-1) \times (n-2) \times \ldots \times 1}
\]
This results in:
\[
\frac{1}{(n+4) \times (n+3) \times (n+2) \times (n+1) \times n}
\]
For educational purposes, it's important to understand the concept of factorials and how they can be expanded and canceled in fractions.
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