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.

Let $a, b, c$ be the sides of a triangle. If $t$ denotes the expression $\frac{\left(a^2+b^2+c^2\right)}{(a b+b c+c a)}$, the set of all possible values of $t$ is

Let $x, y, z$ be three non-negative integers such that $x+y+z=10$. The maximum possible value of $x y z+x y+y z+z x$ is

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 of two numbers be $A$ and geometric mean be $G$, then the numbers will be