Question

If the coefficient of $x^{1274}$ in the expansion of $(x+1)(x-2{)}^2(x+3{)}^3(x-4{)}^4\cdots \cdots (x+49{)}^{49}(x-50{)}^{50}$ is $-k$, then the value of $k$ is

Answer 1

Degree of expansion is
$1+2+3+\cdots +50=\frac{50\times 51}{2}=1275$
$\text{Coefficient of}{x}^{1274}=1-{}^2{C}_1\left(2\right)+{}^3{C}_2\left(3\right)-{}^4{C}_3\left(4\right)+\cdots =1^2-2^2+3^2-4^2+\cdots +{49}^2-{50}^2=-3-7-\dots -99=-(3+7+\dots +99)$
It is a A.P.
$a=3,d=499=3+4(n-1)\Rightarrow n=25$

Therefore, the required coefficient
$=-\frac{25}{2}[3+99]=25\times 51=-1275\therefore k=1275$