Question

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$and common ratio $r$are given.

$a_1=1,r=-\frac{1}{3}$

Short answer

The$nth$term is$a_n={\left\{\frac{-1}{3}\right\}}^{n-1}$and the fifth term is$a_5=\frac{1}{81}$.

Step-by-step solution

Step 1. Given information. 

We have,

$a_1=1$and $r=-\frac{1}{3}$

The $nth$term of a geometric sequence is determined by formula $a_n=a_1{r}^{n-1}$

Step 2. To find the fifth term and the nth term.  

Substitute $a_1=1,r=\frac{1}{3}$and $n=5$

$$\begin{gathered} a_n=1{\left\{-\frac{1}{3}\right\}}^{n-1} \\ {a}_n={\left\{-\frac{1}{3}\right\}}^{n-1} \end{gathered}$$

and

$$\begin{gathered} a_5=1{\left\{-\frac{1}{3}\right\}}^{5-1} \\ {a}_5=1{\left\{-\frac{1}{3}\right\}}^4 \\ {a}_5=\frac{1}{81} \end{gathered}$$