Question

Let $(1 + x)^n = C_o + C_1x + C_2x^2 +........ + C_nx^n$ and $\frac {C_1} {C_0}+ \frac {2.C_2} {C_1} + \frac {3.C_3} {C_2} + ....+ \frac {nC_n}{C_{n-1}} $ =$n \frac {n+1} {K }$ The value of K is

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

Answer 1

Answer: (B)

$\frac{C_{1}}{C_{0}}+2 \frac{C_{2}}{C_{1}}+3 \frac{C_{3}}{C_{2}} + .... \frac{nC_{n}}{C_{n-1}}$
$= \frac{n}{1}+2 \frac{\frac{n\left(n-1\right)}{2}}{n}$
$+3 \frac{\frac{n\left(n-1\right)\left(n-2\right)}{3\,!}}{\frac{n\left(n-1\right)}{2\,!}}+...+\frac{n\cdot1}{n}$
$= n + \left(n - 1\right) + \left(n - 2\right) + .... + 1$
$= \frac{n\left(n+1\right)}{2}$
$\therefore K = 2$.