Question

In a G.P. the $3^{rd}$ term is $24$ and the $6^{th}$ term is $192$. Find the ${10}^{th}$ term.

Answer 1

Step $1:$ Finding first term and common ratio.
We know $n^{th}$ term of G.P. is given by
$a_n=a{r}^{n-1}$
Given: $a_3=24$
$\Rightarrow a{r}^{3-1}=24$
$\Rightarrow a{r}^2=24\cdots \left(i\right)$
and $a_6=192$
$\Rightarrow a{r}^{6-1}=192$
$\Rightarrow a{r}^5=192\cdots \left(ii\right)$
Dividing $\left(ii\right)$ by $\left(i\right)$, we get
$\frac{a{r}^5}{a{r}^2}=\frac{192}{24}$
$\Rightarrow \frac{r^5}{r^2}=8$
$\Rightarrow r^3=8$
$\Rightarrow r^3=2^3$
$\therefore r=2$
Substituting $r=2$ in equation $\left(i\right),$ we get
$a\times 2^2=24$
$\therefore a=6$

Step $2:$ Finding ${10}^{th}$ term.
Substituting $n=10,a=6$and $r=2$ in formula of $n^{th}$ term of G.P ,
$a_n=a{r}^{n-1}$
$\Rightarrow a_{10}=6\times 2^{10-1}$
$\Rightarrow a_{10}=6\times 2^9$
$\therefore a_{10}=3072$