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

For the natural numbers $m, n$,if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots +a_{m+n} y^{m+n}$ and $a_{1}=a_{2}=10$,then the value of $(m+n)$ is equal to:

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

The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}(1+x+x^2)^{2007}$ is equal to

Let $(1 + x + x^2)^{20}(2x + 1) = a_0 + a_1x^1 + a_2x^2 + ... + a_{41}x^{41}$,then $\frac{a_0}{1} + \frac{a_1}{2} + .... + \frac{a_{41}}{42}$ is equal to

Let $C_{r}$ denote the coefficient of $x^{r}$ in the binomial expansion of $(1+x)^{n}$,$n \in N$,$0 \leq r \leq n$. If $P_{n} = C_{0} - C_{1} + \frac{2^{2}}{3}C_{2} - \frac{2^{3}}{4}C_{3} + \dots + \frac{(-2)^{n}}{n+1}C_{n}$,then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.