Problem 81 - Precalculus with Limits

Question

In Exercises 77-84, simplify the factorial expression. \( \dfrac{(n + 1)!}{n!} \)

Step-by-Step Solution

Answer

The simplified form of the factorial expression \( \dfrac{(n + 1)!}{n!} \) is \( n + 1 \).

Step 1 Identify the factorial expression

The given expression is \( \dfrac{(n + 1)!}{n!} \). In this form, it shows the factorial of (n + 1) divided by the factorial of n.

Step 2 Apply the property of factorials

Using the property of factorial where \( (n+1)! = (n+1) * n! \), we can rewrite the expression as \( \dfrac{(n + 1) * n!}{n!} \). The factorials \( n! \) in the numerator and denominator would cancel each other out.

Step 3 Simplify the expression

After cancelling out the factorials, we are left with \( n + 1 \) which is the simplified form of the original expression.

Key Concepts

Factorial Expression

Understanding the basic definition of a factorial expression is the foundation of working with factorials. A factorial of a non-negative integer is the product of all positive integers less than or equal to that integer. This is denoted using the exclamation mark (!). For example, the factorial of 5, written as '5!', would be calculated as 5 x 4 x 3 x 2 x 1, which equals 120.

When looking at factorial expressions in homework problems, we might encounter operations involving factorials, such as the division of factorials, like in the exercise \( \dfrac{(n + 1)!}{n!} \). To approach such problems, it's essential to recognize that the factorial 'grows' each step by the next integer value. So when you encounter a factorial expression, remember it represents a series of multiplications that follows this pattern of growth based on integer values.

Properties of Factorials

The properties of factorials are vital in simplifying complex factorial expressions. One fundamental property is that the factorial of a number 'n' is equal to 'n' times the factorial of 'n - 1'. That is \( n! = n \times (n - 1)! \). This property is recursive and allows for dividing and reducing factorials that appear complex at first sight.

This property was used in the exercise to simplify \( \frac{(n + 1)!}{n!} \). Understanding that \( (n+1)! = (n+1) \times n! \) helps us see how the factorial in the numerator can be expanded and then how parts of the expression can be canceled out when divided by \( n! \), leading to a much simpler expression.

Factorial Division

Factorial division involves dividing one factorial expression by another. To simplify such an expression, you can often reduce the expression by canceling out common terms. The cancellation occurs due to the nature of factorial expressions, where if one factorial is a multiple of the other, the smaller factorial's terms are all factors of the larger factorial.

For example, in the expression \( \dfrac{(n + 1)!}{n!} \), as shown in the exercise, we recognize that \( n! \)