Question

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

$a_1=\sqrt{2},r=\sqrt{2}$

Short answer

The $nth$term is $a_n={\sqrt{2}}^n$and the fifth term is$a_5=4\sqrt{2}$.

Step-by-step solution

Step 1. Given information.  

We have,

$a_1=\sqrt{2},r=\sqrt{2}$

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$a_1=\sqrt{2},r=\sqrt{2}$and$n=5$

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

and

role="math" localid="1647002123469" $$\begin{gathered} a_5=\sqrt{2}{\left(\sqrt{2}\right)}^{5-1} \\ {a}_5=\sqrt{2}{\left(\sqrt{2}\right)}^4 \\ {a}_5=4\sqrt{2} \end{gathered}$$

$a_1=\sqrt{2},r=\sqrt{2}$