If ${a_k} = \frac{1}{{k(k + 1)}},$ for $k = 1,\,2,\,3,\,4,.....,\,n$, then ${\left( {\sum\limits_{k = 1}^n {{a_k}} } \right)^2} = $

$\frac{{{C_1}}}{{{C_0}}} + 2\frac{{{C_2}}}{{{C_1}}} + 3\frac{{{C_3}}}{{{C_2}}} + .... + 15\frac{{{C_{15}}}}{{{C_{14}}}} = $

$\frac{{{C_0}}}{1} + \frac{{{C_1}}}{2} + \frac{{{C_2}}}{3} + .... + \frac{{{C_n}}}{{n + 1}} = $

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

Sum of odd terms is $A$ and sum of even terms is $B$ in the expansion ${(x + a)^n},$ then

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