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=−2(n−1)!

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Identify the general term of the sequence given by the formula \(a_n = -2 (n - 1)!\), where \(n\) is the term number and \((n - 1)!\) denotes the factorial of \((n - 1)\).

Recall that the factorial of a non-negative integer \(k\), written as \(k!\), is the product of all positive integers from 1 up to \(k\). For example, \(0! = 1\), \(1! = 1\), \(2! = 2 \times 1 = 2\), \(3! = 3 \times 2 \times 1 = 6\), and so on.

Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the formula \(a_n = -2 (n - 1)!\):

For each \(n\), compute \((n - 1)!\) and then multiply by \(-2\) to find \(a_n\):

Write down the values of \(a_1, a_2, a_3,\) and \(a_4\) as the first four terms of the sequence.

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주요 개념

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Factorial Notation

The factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 to n. For example, 4! = 4 × 3 × 2 × 1 = 24. By definition, 0! = 1. Factorials grow rapidly and are commonly used in sequences and combinatorics.

General Term of a Sequence

The general term a_n of a sequence defines the nth term as a function of n. It allows you to find any term without listing all previous terms. For example, a_n = -2(n-1)! means each term depends on the factorial of (n-1), scaled by -2.

Evaluating Sequence Terms

To find specific terms of a sequence, substitute the term number n into the general term formula. For a_n = -2(n-1)!, calculate (n-1)! first, then multiply by -2. This process helps write out the first few terms explicitly.

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. an=−2(n−1)!