If the coefficients of $a^{r-1}, a^{r}$ and $a^{r+1}$ in the expansion of $(1+a)^{n}$ are in arithmetic progression, prove that $n^{2}-n(4 r+1)+4 r^{2}-2=0$

If the non zero coefficient of $(2r + 4)th$ term is greater than non zero coefficient of $(r – 2)th$ term in expansion of $(1 + x)^{18}$, then number of possible integral values of $r$ is

The value of $x$, for which the 6th term in the expansion of ${\left\{ {{2^{{{\log }_2}\sqrt {({9^{x – 1}} + 7)} }} + \frac{1}{{{2^{(1/5){{\log }_2}({3^{x – 1}} + 1)}}}}} \right\}^7}$ is $84$, is equal to

For $\mathrm{r}=0,1, \ldots, 10$, let $\mathrm{A}_{\mathrm{r}}, \mathrm{B}_{\mathrm{r}}$ and $\mathrm{C}_{\mathrm{r}}$ denote, respectively, the coefficient of $\mathrm{x}^{\mathrm{r}}$ in the expansions of $(1+\mathrm{x})^{10}$, $(1+\mathrm{x})^{20}$ and $(1+\mathrm{x})^{30}$. Then $\sum_{r=1}^{10} A_r\left(B_{10} B_r-C_{10} A_r\right)$ is equal to

The sum of the rational terms in the binomial expansion of ${\left( {{2^{\frac{1}{2}}} + {3^{\frac{1}{5}}}} \right)^{10}}$ is