Chapter 6, Problem 31CR

To determine

To write:An equation for the nth term of the geometric sequence. Then find $a_9$

Answer to Problem 31CR

An equation for nth term of the sequence is $a_n=486{\left(\frac{1}{3}\right)}^{n-1}$

  $a_9=\frac{2}{27}$

Explanation of Solution

Given information:The sequence $486,162,54,\mathrm{18...}$

Calculation:The first term is 486 and the common ratio is $\frac{1}{3}$ .

  $a_n=a_1{r}^{n-1}$

  $a_n=486{\left(\frac{1}{3}\right)}^{n-1}$

Thus, an equation for nth term of the sequence is $a_n=486{\left(\frac{1}{3}\right)}^{n-1}$

Using the equation to find the 9^th term,

  $\begin{array}{l}{a}_n=486{\left(\frac{1}{3}\right)}^{n-1}\\ \Rightarrow a_9=486{\left(\frac{1}{3}\right)}^{9-1}\\ \Rightarrow a_9=486{\left(\frac{1}{3}\right)}^8\\ \Rightarrow a_9=486\times \frac{1}{6561}\\ \Rightarrow a_9=\frac{2}{27}\end{array}$

Thus, 9^th term of the geometric sequence is $a_9=\frac{2}{27}$