Let $a_{1}, a_{2}, a_{3}, \ldots$ be a G.P. such that $a_{1}<0$; $a_{1}+a_{2}=4$ and $a_{3}+a_{4}=16 .$ If $\sum\limits_{i=1}^{9} a_{i}=4 \lambda,$ then $\lambda$ is equal to 

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 $?$

Consider an infinite $G.P. $ with first term a and common ratio $r$, its sum is $4$ and the second term is $3/4$, then

If $a, b, c$ and $d$ are in $G.P.$ show that:

$\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=(a b+b c+c d)^{2}$

If $G$ be the geometric mean of $x$ and $y$, then $\frac{1}{{{G^2} - {x^2}}} + \frac{1}{{{G^2} - {y^2}}} = $

If $y = x + {x^2} + {x^3} + .......\,\infty ,\,{\rm{then}}\,\,x = $