In Exercises 23–28, evaluate each factorial expression. (2n+1)!/(2n)!

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Identify the factorial expressions in the problem: \((2n+1)!\) and \((2n)!\).

Recall the definition of a factorial: \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\).

Express \((2n+1)!\) in terms of \((2n)!\): \((2n+1)! = (2n+1) \times (2n)!\).

Substitute the expression for \((2n+1)!\) into the original problem: \(\frac{(2n+1) \times (2n)!}{(2n)!}\).

Simplify the expression by canceling out \((2n)!\) from the numerator and the denominator, leaving \(2n+1\).

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Factorial Definition

A factorial, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in permutations, combinations, and various mathematical calculations, making them fundamental in algebra and combinatorics.

Properties of Factorials

Factorials have specific properties that simplify calculations, such as n! = n × (n-1)! and 0! = 1. These properties allow for the reduction of factorial expressions, which is essential when evaluating complex factorial ratios like (2n+1)!/(2n)!. Understanding these properties helps in manipulating and simplifying expressions effectively.

Ratio of Factorials

The ratio of two factorials, such as (2n+1)!/(2n)!, can be simplified by canceling common terms. This involves recognizing that (2n+1)! = (2n+1) × (2n)!, allowing for the expression to reduce to (2n+1). This concept is crucial for evaluating factorial expressions efficiently and accurately.