Chapter 12.3, Problem 23AYU

To determine

To find: The fifth term and $nth$ term of the geometric sequence.

Answer to Problem 23AYU

The fifth term of the geometric sequence is $4\sqrt{2}$ and the $nth$ term is ${\left(\sqrt{2}\right)}^n$ .

Explanation of Solution

Given information:

The first term $a_1=\sqrt{2}$ and the common ratio $r=\sqrt{2}$ .

Calculation:

The formula for nth term of geometric sequence is given by,

  $a_n=a_1{r}^{n-1}$

Calculate the fifth term.

  $\begin{array}{c}{a}_5=\left(\sqrt{2}\right){\left(\sqrt{2}\right)}^{5-1}\\ =\left(\sqrt{2}\right){\left(\sqrt{2}\right)}^4\\ =4\sqrt{2}\end{array}$

Calculate the $nth$ term.

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

Therefore, the fifth term of the geometric sequence is $4\sqrt{2}$ and the $nth$ term is ${\left(\sqrt{2}\right)}^n$ .