Question

In Problems 31–38, find the nth term $a_n$of each geometric sequence. When given, r is the common ratio.

$a_3=\frac{1}{3},a_6=\frac{1}{81}$

Short answer

The common ratio is$r=\frac{1}{3}$and$a_n=3·{\frac{1}{3}}^{n-1}$

Step-by-step solution

Step 1. Write the given information.

Given that$a_3=\frac{1}{3};a_6=\frac{1}{81}$

Step 2. Find the common ratio and nth term.

The formula is

$$\begin{gathered} a_n=a_1·r^{3-1} \\ {a}_3=a_1·r^2 \\ \frac{1}{3}=a_1·r^2 \\ {a}_6=a_1·r^{6-1} \\ \frac{1}{81}=a_1·r^5 \\ \frac{{\displaystyle \frac{1}{3}}}{{\displaystyle \frac{1}{81}}}=\frac{a_1·r^2}{a_1·r^5} \\ 27=\frac{1}{r^3} \\ r=\frac{1}{3} \\ {a}_1=3 \\ {a}_n=3·{\frac{1}{3}}^{n-1} \end{gathered}$$