Question

Să se calculeze suma: ex. A9 - subpunctul f)
Clasa a 10-a , combinări

Answer 1

$\dfrac{1}{k+1}C_{n}^k=\dfrac{n!}{(k+1)k!(n-k)!} =\dfrac{(n+1)!}{(n+1)(k+1)!\Big((n+1)-(k+1)\Big)!}= \\ \\ =\dfrac{1}{n+1}C_{n+1}^{k+1}\\ \\ \\ S_n = \dfrac{C_n^0}{1}+\dfrac{C_n^1}{2}+\dfrac{C_n^2}{3}+...+\dfrac{C_n^n}{n+1}\\\\ S_n = \dfrac{C_{n+1}^{1}}{n+1}+\dfrac{C_{n+1}^2}{n+1}+\dfrac{C_{n+1}^3}{n+1}+...+\dfrac{C_{n+1}^{n+1}}{n+1}\\ \\ (n+1)S_n = C_{n+1}^1+C_{n+2}^2+C_{n+1}^3+...+C_{n+1}^{n+1} \\ \\ (n+1)S_n+C_{n+1}^0 = C_{n+1}^0+C_{n+1}^1+C_{n+1}^2+C_{n+1}^3+...+C_{n+1}^{n+1}$

$(n+1)S_n+1 = 2^{n+1}\\ \\ (n+1)S_n = 2^{n+1}-1 \\ \\ \boxed{S_{n} = \dfrac{2^{n+1}-1}{n+1}}$