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

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.