The sum of infinity of a geometric progression is $\frac{4}{3}$ and the first term is $\frac{3}{4}$. The common ratio is

If the first and the $n^{\text {th }}$ term of a $G.P.$ are $a$ and $b$, respectively, and if $P$ is the product of $n$ terms, prove that $P^{2}=(a b)^{n}$

If ${p^{th}},\;{q^{th}},\;{r^{th}}$ and ${s^{th}}$ terms of an $A.P.$ be in $G.P.$, then $(p - q),\;(q - r),\;(r - s)$ will be in

The number of bacteria in a certain culture doubles every hour. If there were $30$ bacteria present in the culture originally, how many bacteria will be present at the end of $2^{\text {nd }}$ hour, $4^{\text {th }}$ hour and $n^{\text {th }}$ hour $?$

If the range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite $G.P.$, whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$, is equal to...........

The number which should be added to the numbers $2, 14, 62$ so that the resulting numbers may be in $G.P.$, is