If $\sum\limits_{r = 0}^{25} {\left( {^{50}{C_r} \cdot ^{50 - r}{C_{25 - r}}} \right) = K\left( {^{50}{C_{25}}} \right)}$,then $K$ is equal to

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

Let $S_n = 1 + q + q^2 + ..... + q^n$ and $T_n = 1 + \left( \frac{q + 1}{2} \right) + \left( \frac{q + 1}{2} \right)^2 + ...... + \left( \frac{q + 1}{2} \right)^n$ where $q$ is a real number and $q \neq 1$. If $^{101}C_1 + ^{101}C_2 \cdot S_1 + ...... + ^{101}C_{101} \cdot S_{100} = \alpha \cdot T_{100}$,then $\alpha$ is equal to

If $a$ is the arithmetic mean of $b$ and $c$,and $G_1, G_2$ are the two geometric means between them,then $G_1^3 + G_2^3 = $

Find the $n^{th}$ term of the $H.P.$ whose first two terms are $6$ and $3,$ respectively.

Let $b_1, b_2, \dots, b_n$ be a geometric sequence such that $b_1 + b_2 = 1$ and $\sum_{k=1}^{\infty} b_k = 2$. Given that $b_2 < 0$,then the value of $b_1$ is: