Question

$(x+1{)}^n=C_0+C_1{x}^1+C_2{x}^2....C_n{x}^n$.

Find the value of $\frac{C_0}{1}+\frac{C_1}{2}+\frac{C_2}{3}....+\frac{C_n}{n+1}$ :

• A. $2^n$
• B. $3^n$
• C. $\left(\frac{2^{n+1}}{n+1}\right)$
• D. $(\frac{3^{n+1}}{2}-\frac{1}{3})$

Answer 1

The correct option is C $\left(\frac{2^{n+1}}{n+1}\right)$

$(x+1{)}^n=C_0+C_{1}x+C_2{x}^2....................C_n{x}^n$

Integrating w.r.t. to $x$.

$\int (x+1{)}^{n}dx=\int (C_0+C_{1}x+C_2{x}^2....................C_n{x}^n)dx$

$\frac{(x+1{)}^{n+1}}{n+1}=C_{0}x+C_1\frac{x^2}{2}+C_2\frac{x^3}{3}+........+C_n\frac{x^{n+1}}{n+1}$

Now, putting $x=1$

$\Rightarrow \frac{2^{n+1}}{n+1}=\frac{C_0}{1}+\frac{C_1}{2}+\frac{C_2}{3}+..........+\frac{C_n}{n+1}$