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

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Identify the general term of the sequence given by $a_n = \frac{1}{(n - 1)!}$, where $n$ is a positive integer starting from 1.

Recall that the factorial function $k!$ is defined as the product of all positive integers from 1 up to $k$, with the special case $0! = 1$.

Calculate the first four terms by substituting $n = 1, 2, 3, 4$ into the general term formula:

$a_1 = \frac{1}{(1 - 1)!} = \frac{1}{0!}$,

$a_2 = \frac{1}{(2 - 1)!} = \frac{1}{1!}$,

$a_3 = \frac{1}{(3 - 1)!} = \frac{1}{2!}$,

$a_4 = \frac{1}{(4 - 1)!} = \frac{1}{3!}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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