Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. (2n)! / (2n − 1)! = 2n

검증된 단계별 안내

Recall the definition of factorial: for any positive integer $k$, $k! = k \times (k-1) \times (k-2) \times \cdots \times 1$.

Write out the factorial expressions explicitly: $(2n)! = (2n) \times (2n - 1) \times (2n - 2) \times \cdots \times 1$ and $(2n - 1)! = (2n - 1) \times (2n - 2) \times \cdots \times 1$.

Set up the fraction and simplify by canceling common terms: $\frac{(2n)!}{(2n - 1)!} = \frac{(2n) \times (2n - 1) \times \cdots \times 1}{(2n - 1) \times \cdots \times 1} = 2n$.

Conclude that the statement $\frac{(2n)!}{(2n - 1)!} = 2n$ is true because the factorial terms cancel except for the factor \$2n$ in the numerator.

Therefore, the expression simplifies exactly to \$2n$, confirming the statement is correct.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Factorials and Their Properties

A factorial, denoted n!, is the product of all positive integers from 1 up to n. Understanding how factorials expand and simplify is essential, especially when dealing with ratios like (2n)! / (2n − 1)!, where many terms cancel out.

Simplifying Factorial Expressions

When dividing factorials such as (2n)! by (2n − 1)!, most terms cancel, leaving only the highest term, 2n. This simplification helps determine if the given expression equals 2n or not.

Verification Using Counterexamples

To test the truth of a statement involving factorials, substituting specific values of n can confirm or refute it. A single counterexample disproves the statement, while consistent results support its validity.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.b. (2n)! / (2n − 1)! = 2n