Question

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

$-3,-1,-\frac{1}{3},\frac{1}{9},...$

Short answer

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

Step-by-step solution

Step 1. Write the given information.

The given geometric sequence is$-3,1,-\frac{1}{3},-\frac{1}{9},...$

Step 2. Find the common ratio.

$$\begin{gathered} a_1=-3,a_2=1,a_3=-\frac{1}{3} \\ \frac{a_2}{a_1}=-\frac{1}{3} \\ \frac{a_3}{a_2}=\frac{-{\displaystyle \frac{1}{3}}}{1} \\ =-\frac{1}{3} \\ r=-\frac{1}{3} \end{gathered}$$

Step 3. Find the nth term.

The formula for the nth term is $a_n=a_1·r^{n-1}$

$a_n=-3·-{\frac{1}{3}}^{n-1}$