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 = n²/n!

Verified step by step guidance

1. Understand the general term of the sequence given by \(a_n = \frac{n^2}{n!}\), where \(n!\) (n factorial) means the product of all positive integers from 1 up to \(n\).
2. Recall that \(n! = 1 \times 2 \times 3 \times \cdots \times n\), and by definition, \$0! = 1$.
3. Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the formula \(a_n = \frac{n^2}{n!}\):
4. For each \(n\), compute the numerator \(n^2\) and the denominator \(n!\) separately, then divide to find \(a_n\).
5. Write down the terms as \(a_1 = \frac{1^2}{1!}\), \(a_2 = \frac{2^2}{2!}\), \(a_3 = \frac{3^2}{3!}\), and \(a_4 = \frac{4^2}{4!}\), simplifying each fraction as much as possible.

Key Concepts

Sequences and General Terms

A sequence is an ordered list of numbers defined by a general term formula a_n, which gives the nth term. Understanding how to substitute values of n into the formula allows you to find specific terms in the sequence.

Factorials

The factorial of a positive integer n, denoted n!, is the product of all positive integers from 1 to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials grow rapidly and are commonly used in sequences and combinatorics.

Evaluating Terms Involving Factorials

When a sequence term involves factorials, you must carefully compute the factorial values and perform the arithmetic operations as indicated. For a_n = n^2 / n!, calculate n^2 and divide by n! for each term to find the sequence values.