If $(1 + ax)^n = 1 + 8x + 24x^2 + ....,$ then the value of $a$ and $n$ is

If $\sum_{r=0}^{10} \left( \frac{10^{r+1}-1}{10^r} \right) \cdot {}^{11}C_{r+1} = \frac{\alpha^{11}-11^{11}}{10^{10}}$,then $\alpha$ is equal to :

For a real number $r$,let $[r]$ denote the largest integer less than or equal to $r$. Let $a > 1$ be a real number which is not an integer,and let $k$ be the smallest positive integer such that $[a^k] > [a]^k$. Then,which of the following statements is always true?

If $\frac{(1-px)^{-1}}{(1-qx)}=a_0+a_1x+a_2x^2+a_3x^3+\ldots$,then $a_n=$

The value of $(\sqrt{5} + 1)^5 - (\sqrt{5} - 1)^5$ is

The first $3$ terms in the expansion of $(1 + ax)^n$ $(n \ne 0)$ are $1, 6x$ and $16x^2$. Then the value of $a$ and $n$ are respectively