Suppose $\det \left[\begin{array}{cc}\sum _{k=0}^{n}k & \sum _{k=0}^n{}^{n}C_k{k}^2\\ {\sum _{k=0}^n}^n{C}_{k}k& {\sum _{k=0}^n}^n{C}_k{3}^k\end{array}\right]=0$ Holds for some positive integer n. The $\sum _{k=0}^n\frac{{ }^n{C}_k}{k+1}$ equals 

$\begin{array}{l}\left|\begin{array}{cc}\frac{n(n+1)}{2}& n(n-1)2^{n-2}+n{2}^{n-1}\\ n{2}^{n-1}& 4^n\end{array}\right|=0\\ \Rightarrow n\left((n+1)2^{2n-1}-n(n-1)2^{2n-3}-n{2}^{2n-2}\right)=0\\ Cancel out 2^{2n-3} from each term as common term \\ \Rightarrow 4(n+1)-n(n-1)-2n=0\\ \Rightarrow n^2-3n-4=0\Rightarrow n=4(\because n>0,n\in N)\\ \sum _{k=0}^4\frac{{}^{4}C_k}{k+1}=\frac{2^5-1}{5}=\frac{31}{5}=6.20\end{array}$