Let ${T_r}$ be the ${r^{th}}$ term of an $A.P.$ for $r = 1,\;2,\;3,....$. If for some positive integers $m,\;n$ we have ${T_m} = \frac{1}{n}$ and ${T_n} = \frac{1}{m}$, then ${T_{mn}}$ equals

Let $S_{1}$ be the sum of first $2 n$ terms of an arithmetic progression. Let, $S_{2}$ be the sum of first $4n$ terms of the same arithmetic progression. If $\left( S _{2}- S _{1}\right)$ is $1000,$ then the sum of the first $6 n$ terms of the arithmetic progression is equal to:

For $\mathrm{x} \geq 0$, the least value of $\mathrm{K}$, for which $4^{1+\mathrm{x}}+4^{1-\mathrm{x}}$, $\frac{\mathrm{K}}{2}, 16^{\mathrm{x}}+16^{-\mathrm{x}}$ are three consecutive terms of an $A.P.$ is equal to :

If all interior angle of quadrilateral are in $A.P.$ If common difference is $10^o$, then find smallest angle ? .............. $^o$

The sum of first $n$ natural numbers is