Question

If $n>2$ and $\alpha ,\beta ,\gamma$ are three real numbers, then value of
$S=\alpha C_0+(\alpha +\beta )C_1+(\alpha +2\beta +2^2\gamma )C_2-(\alpha +3\beta +3^2\gamma )C_3-...$upto $(n+1)$ terms is

• A. $0$
• B. $2^{n-2}\gamma$
• C. $n^2{2}^{n-2}\gamma$
• D. $n\gamma$

Answer 1

The correct option is D $n\gamma$

$S=\alpha \sum _{k=0}^n{(-1)}^k{C}_k+\beta \sum _{k=1}^n{(-1)}^k\left(k{C}_k\right)+\gamma \sum _{k=2}^n{(-1)}^k\left(k^2{C}_k\right)$

But $\sum _{k=1}^n{(-1)}^k{C}_k=0.$

$\sum _{k=1}^n{(-1)}^k\left(k{C}_k\right)={\left[\frac{d}{dx}{(1-x)}^n\right]}_{x=1}=0$

and $\sum _{k=2}^n{(-1)}^k\left(k^2{C}_k\right)$

$=\sum _{k=2}^n{(-1)}^k\left(k^2{C}_k\right)={\sum }_{k=2}^n{(-1)}^k[k(k-1)+k]C_k$

$=\sum _{k=2}^n{(-1)}^{k}k(k-1)C_k+\sum _{k=1}^n{(-1)}^k\left(k{C}_k\right)+C_1=0+0+n=n$