If $a, b, c$ are in $GP$ and $4a, 5b, 4c$ are in $AP$ such that $a + b + c = 70$,  then value of $a^3 + b^3 + c^3$ is

Let the sum of an infinite $G.P.$, whose first term is $a$ and the common ratio is $r$, be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first $21$ terms of an $AP$, whose first term is $10\,ar , n ^{\text {th }}$ term is $a_{n}$ and the common difference is $10{a r^{2}}$, 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

If the equation $x^8 - kx^2 + 3 = 0$ has a real solution, then least integral value of $k$ is-

The ratio of the $A.M.$ and $G.M.$ of two positive numbers $a$ and $b,$ is $m: n .$ Show that $a: b=(m+\sqrt{m^{2}-n^{2}}):(m-\sqrt{m^{2}-n^{2}})$

If first three terms of sequence $\frac{1}{{16}},a,b,\frac{1}{6}$ are in geometric series and last three terms are in harmonic series, then the value of $a$ and $b$ will be