Question

If $C_0,C_1,C_2....,C_n$ denote the binomial coefficients in the expansion of ${(1+x)}^n$, then $\frac{C1}{C0}+2\frac{C2}{C1}++3\frac{C3}{C2}+.....+n\frac{Cn}{Cn-1}$ equals

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

Answer 1

The correct option is D $\frac{n(n+1)}{2}$

Writing the general term, we get
$T_r=r.\frac{{}^n{C}_r}{{}^n{C}_{r-1}}$
$=r.\frac{\frac{n!}{(n-r)!.r!}}{\frac{n!}{(n-(r-1\left)\right)!.(r-1)!}}$
$=r.\frac{n-(r-1))!.(r-1)!}{(n-r)!.r!}$
$=r.\frac{(n-(r-1)}{r}$
$=(n-(r-1\left)\right)$
$=n+1-r$
Applying summation
${\sum }_{r=1}{r}^n(n+1)-r$
$=n(n+1)-\frac{n(n+1)}{2}$
$=(n+1)(n-\frac{n}{2})$
$=\frac{n(n+1)}{2}$.