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)!

Verified step by step guidance

1. Understand the general term of the sequence given by \(a_n = 2(n+1)!\). Here, \((n+1)!\) means the factorial of \((n+1)\), which is the product of all positive integers from 1 up to \((n+1)\).
2. Recall that the factorial function is defined as \(k! = k \times (k-1) \times (k-2) \times \cdots \times 1\) for any positive integer \(k\), and \$0! = 1$ by definition.
3. To find the first four terms, substitute \(n = 1, 2, 3, 4\) into the formula \(a_n = 2(n+1)!\) one at a time.
4. Calculate each term step-by-step: for each \(n\), first compute \((n+1)!\), then multiply the result by 2 to get \(a_n\).
5. Write down the four terms in order: \(a_1\), \(a_2\), \(a_3\), and \(a_4\) as the first four terms of the sequence.

Key Concepts

Factorials

A factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials grow very quickly 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 in the sequence without listing all previous terms. For example, a_n = 2(n+1)! gives a formula to compute each term directly.

Evaluating Sequence Terms

To find specific terms of a sequence, substitute the term number n into the general term formula and simplify. For instance, to find the first four terms of a_n = 2(n+1)!, calculate a_1, a_2, a_3, and a_4 by plugging in n = 1, 2, 3, and 4.