In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=(n+1)!/n^2

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Identify the general term of the sequence: \(a_n = \frac{(n+1)!}{n^2}\).

To find the first term \(a_1\), substitute \(n = 1\) into the general term: \(a_1 = \frac{(1+1)!}{1^2}\).

To find the second term \(a_2\), substitute \(n = 2\) into the general term: \(a_2 = \frac{(2+1)!}{2^2}\).

To find the third term \(a_3\), substitute \(n = 3\) into the general term: \(a_3 = \frac{(3+1)!}{3^2}\).

To find the fourth term \(a_4\), substitute \(n = 4\) into the general term: \(a_4 = \frac{(4+1)!}{4^2}\).

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주요 개념

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

Factorials

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 grow rapidly and are commonly used in permutations, combinations, and sequences. Understanding how to compute factorials is essential for evaluating expressions involving them.

Sequences

A sequence is an ordered list of numbers defined by a specific rule or formula. Each number in the sequence is called a term, and the position of a term is typically denoted by n. In this case, the sequence is defined by the general term a_n = (n+1)!/n^2, which allows us to calculate specific terms by substituting values for n.

Evaluating Expressions

Evaluating expressions involves substituting values into a mathematical formula to compute specific results. For the sequence given, we need to substitute n = 1, 2, 3, and 4 into the expression a_n = (n+1)!/n^2 to find the first four terms. This process requires careful arithmetic and an understanding of the order of operations.