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