Question

$ \frac {C_1}{C_o} + 2 \frac {C_2}{C_1} +3\frac {C_3}{C_2} + .... +n \frac {C_n}{C_{n-1}} $ =

• A. $ \frac{n(n-1)}{2} $
• B. $ \frac{n(n+1)}{2} $
• C. $ \frac{(n+1)(n+2)}{2} $
• D. $None \,of \,these$

Answer 1

The Correct Option is B

$\frac{C_{1}}{C_{0}}+2 \frac{C_{2}}{C_{1}}+3\frac{C_{3}}{C_{2}} +\ldots+ n-\frac{C_{n}}{C_{n-1}}$
$=\frac{^{n}C_{1}}{^{n}C_{0}}+2 \frac{^{n}C_{2}}{^{n}C_{1}}+3\frac{^{n}C_{3}}{^{n}C_{2}}+\ldots+n \frac{^{n}C_{n}}{^{n}C_{n-1}}$
$=\frac{n}{1}+2\times\frac{\frac{n\left(n-1\right)}{2}}{n}+3\times\frac{\frac{n\left(n-1\right)\left(n-2\right)}{3\times2}}{\frac{n\left(n-1\right)}{2}}+\ldots+n\times\frac{1}{n}$
$=n+\left(n-1\right)+\left(n-2\right)+\ldots+1$
$=\sum n=\frac{n \left(n+1\right)}{2}$