Let $\frac{1}{16}, a$ and $b$ be in $G.P.$ and $\frac{1}{ a }, \frac{1}{ b }, 6$ be in $A.P.,$ where $a , b >0$. Then $72( a + b )$ is equal to ...... .

Given a sequence of $4$ numbers, first three of which are in $G.P.$ and the last three are in $A.P$. with common difference six. If first and last terms in this sequence are equal, then the last term is

If $\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}(x \ne 0)$, then $a,\;b,\;c,\;d$ are in

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a $G.P.$ be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an $A.P.$ If the $5^{\text {th }}$ term of the $G.P.$ is the $\mathrm{n}^{\text {th }}$ term of the $A.P.$, then $\mathrm{n}$ is equal to

Let $a, b$ and $c$ be the $7^{th},\,11^{th}$ and $13^{th}$ terms respectively of a non -constant $A.P.$ If these are also the three consecutive terms of a $G.P.$ then $\frac {a}{c}$ is equal to

Let $f: R \rightarrow R$ be such that for all $\mathrm{x} \in \mathrm{R}\left(2^{1+\mathrm{x}}+2^{1-\mathrm{x}}\right), f(\mathrm{x})$ and $\left(3 ^\mathrm{x}+3^{-\mathrm{x}}\right)$ are in $A.P.$, then the minimum value of $f(x)$ is