In a GP the 3rd term is 24 and the 6th term is 192.

Question:

In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.

Solution:

Let $a$ be the first term and $r$ be the common ratio.

$\therefore a_{3}=24$ and $a_{6}=192$

$\Rightarrow a r^{2}=24$ and $a r^{5}=192$

$\Rightarrow \frac{a r^{5}}{a r^{2}}=\frac{192}{24}$

$\Rightarrow r^{3}=8$

$\Rightarrow r^{3}=2^{3}$

$\Rightarrow r=2$

Putting $r=2$ in $a r^{2}=24$

$a(2)^{2}=24$

$\Rightarrow a=6$

Now, $10^{\text {th }}$ term $=a_{10}=a r^{9}$

Putting $a=6$ and $r=2$ in $a_{10}=a r^{9}$

$\Rightarrow a_{10}=(6)(2)^{9}=3072$

Thus, the $10^{\text {th }}$ term of the G.P. is 3072 .