Question

If $n\ge 2$, value of $S=C_0-3C_1+5C_2-7C_3+....$ upto $(n+1)$ terms is

• A. $0$
• B. ${{}^{2n}C}_n$
• C. ${(-1)}^n$
• D. None of these

Answer 1

The correct option is A $0$

$x(1-x^2{)}^n={}^n{C}_{0}x-{}^n{C}_1{x}^3+{}^n{C}_2{x}^5-...+(-1{)}^n{}^n{C}_n{x}^{2n+1}$
Differentiating with respect to x, we get $(1-x^2{)}^n-nx(1-x^2{)}^{2n-1}={}^n{C}_0-3{}^n{C}_1{x}^2+5{}^n{C}_2{x}^4-...+(-1{)}^n(2n+1){}^n{C}_n{x}^{2n}$
Hence $(1-x^2{)}^n-nx(1-x^2{)}^{2n-1}{|}_{x=1}={}^n{C}_0-3{}^n{C}_1+5{}^n{C}_2-...+(-1{)}^n(2n+1){}^n{C}_n{x}^{2n}$
Hence ${}^n{C}_0-3{}^n{C}_1+5{}^n{C}_2-...+(-1{)}^n(2n+1){}^n{C}_n{x}^{2n}$
$=0$