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Chapter 9: Problem 14

Rewrite as an expression that does not contain factorials. $$\frac{(n+1) !}{(n-1) !}$$

Short Answer

Expert verified
\((n+1) \times n\) is the expression without factorials.

Step by step solution

01

Understand the factorial definition

The factorial of a positive integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \).
02

Identify common terms

Since the expression involves \((n+1)!\) and \((n-1)!\), notice that \((n+1)! = (n+1) \times n \times (n-1)\times (n-2) \times \ldots \times 1\). Whereas \((n-1)! = (n-1) \times (n-2) \times \ldots \times 1\).
03

Simplify the fraction

Substitute the factorial expressions into \(\frac{(n+1)!}{(n-1)!}\):\[\frac{(n+1) \times n \times (n-1)!}{(n-1)!}\]Cancel out the common \((n-1)!\) terms in the numerator and denominator, leaving:\[(n+1) \times n\]
04

Write the final simplified expression

After canceling the common \((n-1)!\), we are left with:\[(n+1) \times n\] This is the expression rewritten without any factorials.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorials
Factorials are a fundamental concept in mathematics, especially in combinatorics and algebra. A factorial, denoted by the symbol "!", is defined as the product of an integer and all the positive integers below it. For instance, the factorial of 5, written as 5!, is calculated as:\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]Factorials are used to compute permutations and combinations, playing a crucial role when we count the number of ways to arrange or choose items. The factorial grows rapidly with larger numbers, making it essential to understand how to manipulate expressions containing factorials, a key skill in simplifying algebraic expressions.
Algebraic Expression
An algebraic expression is a combination of numbers, variables, and operators such as addition, subtraction, multiplication, and division. In the expression \[ \frac{(n+1)!}{(n-1)!} \]we see a typical example involving factorials. Here, the expression is composed of two factorials, and our goal is to transform it into a simpler algebraic form. Understanding how to decompose and rearrange terms in algebraic expressions is crucial for solving problems efficiently. In algebra, expressions like these can represent various scenarios, often simplifying complex problems into manageable parts.
Simplification Process
The simplification process is the method of reducing an algebraic expression to its simplest form. This involves canceling out common factors, reducing complexity, and making expressions easier to work with.To simplify the expression \( \frac{(n+1)!}{(n-1)!} \), we need to:
  • Identify the factorial terms: \((n+1)!\) expands to \((n+1) \times n \times (n-1)!\).
  • Recognize common terms in the numerator and denominator, which in this case is \((n-1)!\).
  • Cancel the common factor \((n-1)!\) from both the numerator and the denominator, simplifying the expression to \((n+1) \times n\).
This process effectively reduces the expression by eliminating the factorial notation, resulting in a more straightforward representation. Simplifying expressions not only aids in performing calculations but also provides clearer insight into the problem structure.

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