Question

$\frac{C_0}{1}+\frac{C_1}{2}+\frac{C_2}{3}+..........+\frac{C_{100}}{101}$ equals ?

• A. $\frac{2^{101}}{101}$
• B. $\frac{2^{101}-1}{101}$
• C. $\frac{3^{101}}{101}$
• D. $\frac{3^{101}-1}{101}$

Answer 1

The correct option is A $\frac{2^{101}}{101}$

$C_0+C_{1/2}+C_{2/3}+.....\frac{C_n}{n+1}$

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

$\int (1+x{)}^n=\int x_0+C_{1}x.......C_n{x}^n$

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

$x=1$

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

$=\frac{(2{)}^{101}}{101}$