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^2/n!

Verified step by step guidance

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

Key Concepts

Sequences

A sequence is an ordered list of numbers that follow a specific rule or pattern. Each number in the sequence is called a term, and the position of each term is typically denoted by an index, such as 'n'. Understanding how to identify and generate terms in a sequence is crucial for solving problems related to sequences.

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 very quickly and are often used in combinatorial problems and sequences. Recognizing how to compute factorials is essential for evaluating terms in sequences that involve them.

General Term of a Sequence

The general term of a sequence is a formula that allows you to calculate any term in the sequence based on its position, n. In this case, the general term is given as a_n = n^2/n!. Understanding how to manipulate and evaluate this formula is key to finding specific terms in the sequence, such as the first four terms.