Question

Find the geometric series whose $5^{\mathrm{th}}$ and $8^{\mathrm{th}}$ terms are $80$ and $640$, respectively.

Answer 1

Step 1. Find the common ratio:

Here it is given that $5^{\mathrm{th}}$ term is $80$ and $8^{\mathrm{th}}$ term is $640$

Let the first term is $a$ and the common ratio is$\mathrm{r}$

${\mathrm{a}}_5={\mathrm{ar}}^4=80....\left(\mathrm{i}\right)$

${\mathrm{a}}_8={\mathrm{ar}}^7=640....\left(\mathrm{ii}\right)$

Divide (ii) and (i)

$\begin{array}{rcl}& & \begin{array}{cc}\Rightarrow & \frac{{\mathrm{ar}}^7}{{\mathrm{ar}}^4}=\frac{640}{80}\\ \Rightarrow & r^3=8\\ \Rightarrow & r=2\end{array}\end{array}$

Step 2. Find the GP series:

Put the value of $2$ in eq $\left(i\right)$

$\begin{array}{rcl}& & \begin{array}{cc}\Rightarrow & {\mathrm{ar}}^4=80\\ \Rightarrow & a\times 2^4=80\\ \Rightarrow & a=\frac{80}{16}\\ \Rightarrow & a=5\end{array}\end{array}$

Second term,

$\Rightarrow \mathrm{ar}=5\times 2=10$

Third term

$\Rightarrow {\mathrm{ar}}^2=5\times 2^2=20$

Fourth term

$\Rightarrow {\mathrm{ar}}^3=5\times 2^3=40$

Therefore,the required GP is 5,10,20,40,80,..640.