Question

$\frac{C_1}{C_0}+2.\frac{C_2}{C_1}+3.\frac{C_3}{C_2}+.....+n.\frac{C_n}{C_{n-1}}$=

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

Answer 1

Answer: (A)

$\frac{n(n+1)}{2}$

$\frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{n-(r-1)}{r}$

$\Rightarrow \frac{{}^n{C}_{r+1}}{{}^n{C}_r}=\frac{n-r}{r+1}$

Tr for the given sequence is

$Tr={\sum }_{r=0}^{n-1}(r+1)\left(\frac{n-r}{r+1}\right)$

$={\sum }_{r=0}^{n-1}n-r={\sum }_{r=0}^{n-1}n-{\sum }_{r=0}^{n-1}r$

$=n(n+1)-\frac{(n+1)\left(n\right)}{2}$

$=\frac{n(n+1)}{2}$