Question

Let $S=C_0^2+1.\frac{C_1^2}{C_0}+2.\frac{C_2^2}{C_1}+3.\frac{C_3^2}{C_2}+.........+n.\frac{C_n^2}{C_{n-1}}$ Where $C_r={}^n{C}_r$ then which of the following is wrong

• A. 2ⁿ divides 'S'
• B. (n+2) divides (S+n)
• C. 2^n-1 divides (S+n)
• D. 'S' is a positive integer

Answer 1

The correct option is A

2ⁿ divides 'S'

$\frac{r.{\left({}^n{C}_r\right)}^2}{{}^n{C}_{r-1}}=(n-r+1)C_r$ G.E = ${\sum }_{r=1}^n\left(\right(n+1)-r)C_r=$

$(n+1)(2^n-1)-n{.2}^{n-1}=(n+2)2^{n-1}-(n+1)$

$S=C_0^2+{\sum }_{r=1}^n\frac{r.C_r^2}{C_{r-1}}=(n+2){.2}^{n-1}-n\Rightarrow S+n=(n+2){.2}^{n-1}$

Which is divisible by 'n + 2' and $2^{n-1}$ & also 'S' is positive integer