Question

The value of $C_0{+}^3{C}_1{+}^5{C}_2{+}^7{C}_3+.............{+}^{(2n+1)}{C}_n$ is equal to

• A. $2^n$
• B. $2^n+n{.2}^{n-1}$
• C. $2^n.(n+1)$
• D. none of these

Answer 1

The correct option is C $2^n.(n+1)$

Consider the following
$(1+x^2{)}^n=1+{}^n{C}_1{x}^2+{}^n{C}_2{x}^4+{}^n{C}_3{x}^6...{}^n{C}_n{x}^{2n}$
Multiplying both sides with x, we get
$x(1+x^2{)}^n=x+{}^n{C}_1{x}^3+{}^n{C}_2{x}^5+{}^n{C}_3{x}^7...{}^n{C}_n{x}^{2n+1}$
Differentiating both sides with respect to x, we get
$\left[\right(1+x^2{)}^n+nx\left(2x\right)(1+x^2{)}^{n-1}{]}_{x=1}=1+3{}^n{C}_1+5{}^n{C}_2+...(2n+1){}^n{C}_n$
Hence
$1+3{}^n{C}_1+5{}^n{C}_2+...(2n+1){}^n{C}_n$
$=\left[\right(1+x^2{)}^n+nx\left(2x\right)(1+x^2{)}^{n-1}{]}_{x=1}$
$=2^n+2n\left(2^{n-1}\right)$
$=2^n+n{2}^n$
$=2^n(n+1)$