Question

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

$a_1=-2,r=4$

Short answer

The $nth$term is $-{\left(2\right)}^{2n-1}$and the fifth term is$-512$.

Step-by-step solution

Step 1. Given information. 

We have, $a_1=-2$and $r=4$

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 $a_1=-2,r=4$and $n=5$

$$\begin{gathered} a_n=-2\times 4^{n-1} \\ {a}_n=-2\times {\left(2^2\right)}^{n-1}{a}_n=-2^{1+2n-2} \\ {a}_n=-2^{2n-1} \end{gathered}$$

and

$$\begin{gathered} a_5=-2\times 4^{5-1} \\ {a}_5=-2\times 4^4 \\ {a}_5=-512 \end{gathered}$$