The sum of the coefficients in the expansion of $(1+x+x^2)^n$ is

If $1 + x^4 + x^5 = \sum\limits_{i = 0}^5 a_i (1 + x)^i$ for all $x$ in $\mathbb{R},$ then $a_2$ is

If $n$ is a positive integer,then $(\sqrt{3} + 1)^{2n} - (\sqrt{3} - 1)^{2n}$ is

Find an approximation of $(0.99)^{5}$ using the first three terms of its binomial expansion.

For $|x| < \frac{1}{5}$,the coefficient of $x^3$ in the expansion of $\frac{1}{(1-5 x)(1-4 x)}$ is

The expression $\frac{1}{{\sqrt {4x + 1} }}\left[ {{{\left[ {\frac{{1 + \sqrt {4x + 1} }}{2}} \right]}^7} - {{\left[ {\frac{{1 - \sqrt {4x + 1} }}{2}} \right]}^7}} \right]$ is a polynomial in $x$ of degree