Question:
Find the $6^{\text {th }}$ and $n$th terms of the GP $2,6,18,54 \ldots$
Solution:
Given: GP is 2, 6, 18, 54….
The given GP is of the form, $a, a r, a r^{2}, a r^{3} \ldots$
Where r is the common ratio.
First term in the given GP, $a_{1}=a=2$
Second term in GP, $a_{2}=6$
Now, the common ratio, $r=\frac{a_{2}}{a_{1}}$
$r=\frac{6}{2}=3$
Now, $\mathrm{n}^{\text {th }}$ term of GP is, $a_{n}=a r^{n-1}$
So, the $6^{\text {th }}$ term in the GP,
$\mathrm{a}_{6}=\mathrm{ar}^{5}$
$=2 \times 3^{5}$
$=486$
$\mathrm{n}^{\text {th }}$ term in the GP,
$a_{n}=a r^{n-1}$
$=2.3^{n-1}$
Hence, $6^{\text {th }}$ term $=486$ and $n^{\text {th }}$ term $=2.3^{n-1}$