Let $a$ be the first term and $r$ be the common ratio of a GP.
$\therefore P$ th, Qth and R th terms of a GP are respectively $a r^{P-1}, a r^{Q-1}$ and $a r^{R-1}$.
According to question,
$a r^{P-1}=64$ ...(i)
$a r^{Q-1}=27$ ...(ii)
$a r^{R-1}=36$ ...(iii)
Dividing Eq. (i) by Eq. (ii), we get
$r^{P-Q}=\left(\frac{4}{3}\right)^{3}$
Dividing Eq. (ii) by Eq. (iii), we get
$r^{Q-R}=\frac{3}{4}$ ...(iv)
Dividing Eq. (ii) by Eq. (iii), we get
$r^{Q-R}=\frac{3}{4}$
$\Rightarrow r^{3 Q-3 R}=\left(\frac{3}{4}\right)^{3}$ ...(v)
Multiplying Eq. (iv) and Eq. (v), we get
$r^{P-Q} \times r^{3 Q-3 R} =1$
$\Rightarrow r^{P-Q+3 Q-3 R} =1$
$\Rightarrow r^{P+2 Q-3 R} =r^{0}$
$\Rightarrow P+2 Q-3 R =0$
$\Rightarrow P+2 Q =3 R$