Question

If $n>2$ and $\alpha ,\beta ,\gamma \in R$ , then the value of $S=\alpha C_0-\left(\alpha +\beta \right)C_1$ $+\left(\alpha +2\beta +2^2\gamma \right)C_2$ $-\left(\alpha +3\beta +3^2\gamma \right)C_3+....$ upto $\left(n+1\right)$ terms is equal to

(where, $C_r$ denotes ${}^n{C}_r$ )

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

Answer 1

Answer: (D)

$S=\alpha \left(C_0-C_1+C_2-C_3\dots \dots \right)+\beta \left(-C_1+2C_2-3C_3+\dots \dots \right)$
$+\gamma \left(2^2{C}_2-3^2{C}_3+\dots \dots \right)$
$=\alpha {\sum }_{k=0}^n(-1{)}^k\left(C_k\right)+\beta {\sum }_{k=1}^n(-1{)}^k\left(k{C}_k\right)+\gamma {\sum }_{k=2}^n(-1{)}^k\left(k^2{C}_k\right)$
But, ${\sum }_{k=0}^n(-1{)}^k{C}_k=0$
${\sum }_{k=1}^n(-1{)}^{k}k{C}_k={\frac{d}{dx}(1-x{)}^n|}_{x=1}=0$
and ${\sum }_{k=2}^n(-1{)}^k{k}^2{C}_k={\sum }_{k=2}^n(-1{)}^k\{k(k-1)+k\}{C}_k$
$={\sum }_{k=2}^n(-1{)}^{k}k(k-1)C_k+{\sum }_{k=1}^n(-1{)}^{k}k{C}_k+C_1$
$=0+0+n=n$
$\Rightarrow S=\alpha (0)+\beta (0)+\gamma (n)=n\gamma$