If five $G.M.’s$ are inserted between $486$ and $2/3$ then fourth $G.M.$ will be

If $a,\;b,\;c,\;d$ and $p$ are different real numbers such that $({a^2} + {b^2} + {c^2}){p^2} - 2(ab + bc + cd)p + ({b^2} + {c^2} + {d^2}) \le 0$, then $a,\;b,\;c,\;d$ are in

Let the positive numbers $a _1, a _2, a _3, a _4$ and $a _5$ be in a G.P. Let their mean and variance be $\frac{31}{10}$ and $\frac{ m }{ n }$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$, then $m + n$ is equal to $.........$.

The sum of few terms of any ratio series is $728$, if common ratio is $3$ and last term is $486$, then first term of series will be

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...........

There are two such pairs of non-zero real valuesof $a$ and $b$ i.e. $(a_1,b_1)$ and $(a_2,b_2)$ for which $2a+b,a-b,a+3b$ are three consecutive terms of a $G.P.$, then the value of $2(a_1b_2 + a_2b_1) + 9a_1a_2$ is-