\(\frac{C_n^k}{\left(k+1\right)\left(k+2\right)}=\frac{n!}{k!\left(k+1\right)\left(k+2\right)\left(n-k\right)!}=\frac{\left(n+2\right)!}{\left(k+2\right)!\left(n+1\right)\left(n+2\right)\left(n-k\right)!}=\frac{C_{n+2}^{k+2}}{\left(n+1\right)\left(n+2\right)}\)
\(\Rightarrow\sum\limits^n_{k=0}\frac{C_n^k}{\left(k+1\right)\left(k+2\right)}=\sum\limits^n_{k=0}\frac{C_{n+2}^{k+2}}{\left(n+1\right)\left(n+2\right)}=\frac{\sum\limits^n_{k=0}C_{n+2}^{k+2}}{\left(n+1\right)\left(n+2\right)}\) (1)
Xét khai triển:
\(\left(1+x\right)^{n+2}=\sum\limits^{n+2}_{k=0}C_{n+2}^kx^k\)
Thay \(x=1\Rightarrow2^{n+2}=\sum\limits^{n+2}_{k=0}C_{n+2}^k=\sum\limits^n_{k=0}C_{n+2}^{k+2}+C_{n+2}^0+C_{n+2}^1\) (1)
(1);(2) \(\Rightarrow2^{100}-n-3+C_{n+2}^0+C_{n+2}^1=2^{n+2}\)
\(\Leftrightarrow2^{n+2}=2^{100}\Rightarrow n=98\)
