Chapter 12.3, Problem 36E

To determine

Find the common ratio $r$ , fifth term $a_5$ and the $nth$ term $a_n$ of the given geometric sequence $5,5^{c+1},5^{2c+1},5^{3c+1},\mathrm{...............}$

Answer to Problem 36E

Common ratio $r=5^c$ fifth term $a_5=5^{4c+1}$ and the $nth$ term $a_n=5^{c\left(n-1\right)+1}$

Explanation of Solution

Given information:

It is given in the question that, the geometric sequence is $5,5^{c+1},5^{2c+1},5^{3c+1},\mathrm{...............}$

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 $5,5^{c+1},5^{2c+1},5^{3c+1},\mathrm{...............}$ means,

  $\begin{array}{l}{a}_1=5\Rightarrow a=5\\ a_2=5^{c+1}\Rightarrow a{r}^1=5^{c+1}\Rightarrow r=\frac{5^{c+1}}{5}\Rightarrow r=5^{c+1-1}\Rightarrow r=5^c\end{array}$

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

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

And, by putting $a=5,r=5^c$ in $a_n=a{r}^{n-1}$

  $\begin{array}{l}{a}_n=5*{\left(5^c\right)}^{n-1}\\ a_n=5^{c\left(n-1\right)+1}\end{array}$