Evaluate each factorial expression. (n+2)!/n!

Verified step by step guidance

1. Recall the definition of factorial: for any positive integer \(k\), \(k! = k \times (k-1) \times (k-2) \times \cdots \times 1\).
2. Write out the factorial expressions explicitly: \((n+2)! = (n+2) \times (n+1) \times n!\).
3. Substitute this expression into the given fraction: \(\frac{(n+2)!}{n!} = \frac{(n+2) \times (n+1) \times n!}{n!}\).
4. Cancel the common \(n!\) terms in the numerator and denominator, leaving \(\frac{(n+2) \times (n+1) \times \cancel{n!}}{\cancel{n!}} = (n+2)(n+1)\).
5. Express the final simplified form as the product of two binomials: \((n+2)(n+1)\), which can be expanded if needed.

Key Concepts

Factorial Notation

Factorial notation, denoted by n!, represents the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. It is commonly used in permutations, combinations, and algebraic expressions involving sequences.

Simplifying Factorial Expressions

When simplifying expressions involving factorials, such as (n+2)!/n!, it helps to expand the factorial terms to cancel common factors. For instance, (n+2)! = (n+2)(n+1)n!, so dividing by n! leaves (n+2)(n+1). This technique reduces complex factorial expressions to simpler polynomial forms.

Algebraic Manipulation

Algebraic manipulation involves applying arithmetic operations and properties of expressions to simplify or solve problems. In factorial expressions, recognizing patterns and factoring common terms allows for efficient simplification and evaluation, which is essential for solving factorial-related exercises.