Problem 19 - College Algebra

Question

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$18,6,2, \frac{2}{3}, \dots$$

Step-by-Step Solution

Answer

The general term of the sequence is given by \(a_{n} = 18 * (\frac{1}{3})^{(n-1)}\). The seventh term of this sequence is \(\frac{2}{243}\).

Step 1: Calculate the common ratio

Given the sequence \(18, 6, 2, \frac{2}{3}, \ldots\) the ratio between each term is calculated as the second term divided by the first term, or \(r = \frac{b}{a}\). Hence the common ratio \(r\) is \(\frac{6}{18} = \frac{1}{3}\).

Step 2: Derive the general formula

The general formula for a geometric sequence is given by \(a_{n} = a_{1} * r^{(n-1)}\), where \(a_{1}\) is the first term and \(r\) is the common ratio. Plugging our values in, we get \(a_{n} = 18 * (\frac{1}{3})^{(n-1)}\)

Step 3: Find the seventh term

To find the seventh term of a geometric sequence, all that has to be done is to replace \(n\) with 7 in our derived formula. Thus, \(a_{7} = 18 * (\frac{1}{3})^{(7-1)} = 18 * (\frac{1}{3})^{6} = \frac{2}{243}\).