The sum of the series $\sum\limits_{r = 0}^n {(-1)^r \, ^nC_r \left( \frac{1}{2^r} + \frac{3^r}{2^{2r}} + \frac{7^r}{2^{3r}} + \frac{15^r}{2^{4r}} + \dots + m \text{ terms} \right)}$ is

Let $(1+x+x^2)^9=a_0+a_1 x+a_2 x^2 +\ldots+a_{18} x^{18}$. Then

If $C_0, C_1, C_2, \ldots, C_{n}$ are the binomial coefficients in the expansion of $(1+x)^{n}$,then $(C_0+C_1)-(C_2+C_3)+(C_4+C_5)-(C_6+C_7)+\ldots=$

The fractional part of a real number $x$ is defined as $x - [x]$,where $[x]$ is the greatest integer less than or equal to $x$. Let $F_1$ and $F_2$ be the fractional parts of $(44 - \sqrt{2017})^{2017}$ and $(44 + \sqrt{2017})^{2017}$,respectively. Then,$F_1 + F_2$ lies between the numbers:

Let the sixth term in the binomial expansion of $\left(\sqrt{2^{\log_2(10-3^x)}} + \sqrt[5]{2^{(x-2)\log_2 3}}\right)^m$,in the increasing powers of $2^{(x-2)\log_2 3}$,be $21$. If the binomial coefficients of the second,third,and fourth terms in the expansion are respectively the first,third,and fifth terms of an $A.P.$,then the sum of the squares of all possible values of $x$ is $.........$.

Let $R=(5 \sqrt{5}+11)^{2 n+1}$ and $f=R-[R]$,where $[x]$ denotes the greatest integer less than or equal to $x$,then $R f=$