The fourth term of a G.P. is 27 and the 7th term is 729,
Question:

The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.

Solution:

Let $a$ be the first term and $r$ be the common ratio of the given G.P.

$\therefore a_{4}=27$ and $a_{7}=729$

$\Rightarrow a r^{3}=27$ and $a r^{6}=729$

$\Rightarrow \frac{a r^{6}}{a r^{3}}=\frac{729}{27}$

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

$\Rightarrow r=3$

Putting $r=3$ in $a r^{3}=27$

$a(3)^{3}=27$

$\Rightarrow a=1$

Thus, the given G.P. is $1,3,9, \ldots$