Let $a, b, c, d\, \in \, R^+$ and $256\, abcd \geq  (a+b+c+d)^4$ and $3a + b + 2c + 5d = 11$ then $a^3 + b + c^2 + 5d$ is equal to :-

Two sequences $\{ {t_n}\} $ and $\{ {s_n}\} $ are defined by ${t_n} = \log \left( {\frac{{{5^{n + 1}}}}{{{3^{n - 1}}}}} \right)\,,\,\,{s_n} = {\left[ {\log \left( {\frac{5}{3}} \right)} \right]^n}$, then

Let $a, b$ and $c$ be in $G.P$ with common ratio $r,$ where $a \ne 0$ and $0\, < \,r\, \le \,\frac{1}{2}$.  If $3a, 7b$ and $15c$ are the first three terms of an $A.P.,$ then the $4^{th}$ term of this $A.P$ is

If the arithmetic mean and geometric mean of the $p ^{\text {th }}$ and $q ^{\text {th }}$ terms of the sequence $-16,8,-4,2, \ldots$ satisfy the equation $4 x^{2}-9 x+5=0,$ then $p+q$ is equal to ..... .

If $a,\,b,\;c$ are in $A.P.$ and ${a^2},\;{b^2},\;{c^2}$ are in $H.P.$, then

Suppose $a,\,b,\,c$ are in $A.P.$ and ${a^2},{b^2},{c^2}$ are in $G.P.$ If $a < b < c$ and $a + b + c = \frac{3}{2}$, then the value of $a$ is