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

Verified step by step guidance

1. Identify the general term of the sequence: \(a_n = \frac{(n+1)!}{n^2}\).
2. To find the first term \(a_1\), substitute \(n = 1\) into the general term: \(a_1 = \frac{(1+1)!}{1^2}\).
3. To find the second term \(a_2\), substitute \(n = 2\) into the general term: \(a_2 = \frac{(2+1)!}{2^2}\).
4. To find the third term \(a_3\), substitute \(n = 3\) into the general term: \(a_3 = \frac{(3+1)!}{3^2}\).
5. To find the fourth term \(a_4\), substitute \(n = 4\) into the general term: \(a_4 = \frac{(4+1)!}{4^2}\).

Key Concepts

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.