Let $\lambda$ be the positive root of the equation $x^2-x-1=0$,and set $a_n = \frac{1}{\sqrt{5}}\left(\lambda^n - (1-\lambda)^n\right)$ for $n \in N$,where $N$ is the set of all natural numbers. Consider the sets $A = \{ n \in N : a_n \text{ is a rational number, but not an integer} \}$ and $B = \{ n \in N : a_n \text{ is an irrational number} \}$. Then:

Find the coefficient of $a^{4}$ in the product $(1+2a)^{4}(2-a)^{5}$ using the binomial theorem.

For non-negative integers $s$ and $r$,let $\binom{s}{r} = \begin{cases} \frac{s!}{r!(s-r)!} & \text{if } r \leq s \\ 0 & \text{if } r > s \end{cases}$. For positive integers $m$ and $n$,let $g(m, n) = \sum_{p=0}^{m+n} \frac{f(m, n, p)}{\binom{n+p}{p}}$,where for any non-negative integer $p$,$f(m, n, p) = \sum_{i=0}^{p} \binom{m}{i} \binom{n+i}{p} \binom{p+n}{p-i}$. Then which of the following statements is/are $TRUE$?
$(A)$ $g(m, n) = g(n, m)$ for all positive integers $m, n$
$(B)$ $g(m, n+1) = g(m+1, n)$ for all positive integers $m, n$
$(C)$ $g(2m, 2n) = 2g(m, n)$ for all positive integers $m, n$
$(D)$ $g(2m, 2n) = (g(m, n))^2$ for all positive integers $m, n$

Let $R=(5 \sqrt{5}+11)^{2 n+1}$ and $f=R-[R]$,where $[x]$ denotes the greatest integer less than or equal to $x$,then $R f=$

If $a_r$ is the coefficient of $x^r$ in the expansion of $(1 + x + x^2)^n$,then $a_1 - 2a_2 + 3a_3 - \dots - 2n\,a_{2n} = $

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