The positive integer just greater than $(1 + 0.0001)^{10000}$ is

If the sum of the coefficients in the expansion of $(1 - 3x + 10x^2)^n$ is $a$ and if the sum of the coefficients in the expansion of $(1 + x^2)^n$ is $b$,then:

For $|x| < \frac{1}{\sqrt{2}}$, the coefficient of $x$ in the expansion of $\frac{(1 - 4x)^2 (1 - 2x^2)^{1/2}}{(4 - x)^{3/2}}$ is

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?

Prove that $\sum\limits_{r = 0}^n {3^r \,^nC_r} = 4^n$.

Let $[x]$ denote the greatest integer less than or equal to $x$. If for $n \in N$,$(1-x+x^3)^n = \sum_{j=0}^{3n} a_j x^j$,then $\sum_{j=0}^{[\frac{3n}{2}]} a_{2j} + 4 \sum_{j=0}^{[\frac{3n-1}{2}]} a_{2j+1}$ is equal to