Chapter 12.3, Problem 18AYU

To determine

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

Answer to Problem 18AYU

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

Explanation of Solution

Given information:

The first term $a_1=-2$ and the common ratio $r=4$ .

Calculation:

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

  $a_n=a_1{r}^{n-1}$

Calculate the fifth term.

  $\begin{array}{c}{a}_n=\left(-2\right)4^{5-1}\\ =\left(-2\right){\left(4\right)}^4\\ =-512\end{array}$

Calculate $nth$ term.

  $\begin{array}{l}{a}_n=\left(-2\right){\left(4\right)}^{n-1}\\ =\left(-1\right)2^{2n-1}\\ =-{\left(2\right)}^{2n-1}\end{array}$

Therefore, the fifth term of the geometric sequence is $-512$ and the $nth$ term is $-{\left(2\right)}^{2n-1}$ .