Chapter 12.3, Problem 29E

To determine

Find the common ratio $r$ , fifth term $a_5$ and the $nth$ term $a_n$ of the given geometric sequence $0.36,-0.09,0.027,-0.0081,\mathrm{.......}$

Answer to Problem 29E

Common ratio $r=-0.3$ fifth term $a_5=0.00243$ and the $nth$ term $a_n=0.3*{\left(-0.3\right)}^{n-1}$

Explanation of Solution

Given information:

It is given in the question that, the geometric sequence is $0.3,-0.09,0.027,-0.0081,\mathrm{.......}$

Formula used:

  $a_n=a{r}^{n-1}$

Where,

  $\begin{array}{l}aisthe{1}^{st}term\\ nisthenumberofterm\\ risthecommonratio\end{array}$

Calculation:

The geometric sequence is $0.3,-0.09,0.027,-0.0081,\mathrm{.......}$ means,

  $\begin{array}{l}{a}_1=0.3\Rightarrow a=0.3\\ a_2=-0.09\Rightarrow a{r}^1=-0.09\Rightarrow r=\frac{-0.09}{a}\Rightarrow r=\frac{-0.09}{0.3}\Rightarrow r=-0.3\end{array}$

By putting $a=0.3,r=-0.3\&n=5$ in $a_n=a{r}^{n-1}$

  $\begin{array}{l}{a}_5=0.3{\left(-0.3\right)}^{5-1}\\ =0.3{\left(-0.3\right)}^4\\ =0.3*0.0081\\ =0.00243\end{array}$

And, by putting $a=0.3,r=-0.3$ in $a_n=a{r}^{n-1}$

  $a_n=0.3*{\left(-0.3\right)}^{n-1}$