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. Identify the general term of the sequence: \( a_n = 2(n+1)! \).
2. To find the first term \( a_1 \), substitute \( n = 1 \) into the general term: \( a_1 = 2(1+1)! \).
3. To find the second term \( a_2 \), substitute \( n = 2 \) into the general term: \( a_2 = 2(2+1)! \).
4. To find the third term \( a_3 \), substitute \( n = 3 \) into the general term: \( a_3 = 2(3+1)! \).
5. To find the fourth term \( a_4 \), substitute \( n = 4 \) into the general term: \( a_4 = 2(4+1)! \).

Key Concepts

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 rapidly and are fundamental in combinatorics, probability, and sequences. Understanding how to compute factorials is essential for evaluating terms in sequences that involve them.

Sequences

A sequence is an ordered list of numbers that follows a specific rule or pattern. Each number in the sequence is called a term, and the position of a term is typically denoted by n. In this case, the sequence is defined by the general term a_n = 2(n+1)!. Recognizing how to derive individual terms from a general formula is crucial for solving problems related to sequences.

Evaluating Terms

Evaluating terms in a sequence involves substituting specific values of n into the general term formula to find corresponding sequence values. For the given sequence a_n = 2(n+1)!, one would calculate a_1, a_2, a_3, and a_4 by substituting n = 1, 2, 3, and 4 respectively. This process is vital for generating the first few terms of the sequence.