Question

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+...+C_n{x}^n$ then $\frac{C_0}{1}+\frac{C_2}{3}+\frac{C_4}{5}+....=$

• A. $2^n$
• B. $\frac{2^n}{n-1}$
• C. $\frac{2^n}{n+1}$
• D. $\frac{2^n}{n}$

Answer 1

Answer: (C)

$\frac{2^n}{n+1}$

The above expression is nothing but
$\frac{1}{2}{\int }_{x=0}^{x=1}\left[\right(1+x{)}^n+(1-x{)}^n].dx$
$=\frac{1}{2}[\frac{(1+x{)}^{n+1}-(1-x{)}^{n+1}}{n+1}{]}_{x=0}^{x=1}$
$=\frac{1}{2(n+1)}[2^{n+1}+0-(1^{n+1}-1^{n+1}\left)\right]$
$=\frac{2^n}{n+1}$