Problem 22 - College Algebra

Question

The general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$a_{n}=-2(n-1) !$$

Step-by-Step Solution

Answer

The first four terms of the sequence are -2, -2, -4, and -12.

Step 1: Understand the Factorial Operation

A factorial of a positive integer \( n \), denoted by \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 \). Also, \( 0! \) is defined as 1.

Step 2: Deduce the First Term

The first term is obtained by substituting \( n = 1 \) into the equation for the general term. That is \( a_1 = -2(1-1)! = -2 \times 0! = -2 \times 1 = -2 \).

Step 3: Deduce the Second Term

The second term is obtained by substituting \( n = 2 \) into the equation for the general term. That is \( a_2 = -2(2-1)! = -2 \times 1! = -2 \times 1 = -2 \).

Step 4: Deduce the Third Term

The third term is obtained by substituting \( n = 3 \) into the equation for the general term. That is \( a_3 = -2(3-1)! = -2 \times 2! = -2 \times 2 = -4 \).

Step 5: Deduce the Fourth Term

The fourth term is obtained by substituting \( n = 4 \) into the equation for the general term. That is \( a_4 = -2(4-1)! = -2 \times 3! = -2 \times 6 = -12 \).