Write the first four terms of each sequence whose general term is given. $a_n=1/(n-1)!$

Verified step by step guidance

1. Identify the general term of the sequence given by \(a_n = \frac{1}{(n - 1)!}\), where \(n\) is a positive integer starting from 1.
2. Recall that the factorial function \(k!\) is defined as the product of all positive integers from 1 up to \(k\), with the special case \$0! = 1$.
3. Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the general term formula:
4. \(a_1 = \frac{1}{(1 - 1)!} = \frac{1}{0!}\),
5. \(a_2 = \frac{1}{(2 - 1)!} = \frac{1}{1!}\),
6. \(a_3 = \frac{1}{(3 - 1)!} = \frac{1}{2!}\),
7. \(a_4 = \frac{1}{(4 - 1)!} = \frac{1}{3!}\).

Key Concepts

Sequences and Terms

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 indicated by its index n. Understanding how to find terms from a general formula is essential for working with sequences.

Factorials

The factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 to n. By definition, 0! = 1. Factorials grow very quickly and are commonly used in sequences, permutations, and combinations.

Evaluating Terms Using the General Term

To find specific terms of a sequence, substitute the term number n into the general term formula. For example, for a_n = 1/(n - 1)!, calculate the factorial of (n - 1) and then find its reciprocal. This process helps generate the sequence's terms explicitly.