Suppose, $\operatorname{det}\left[\begin{array}{cc}\sum_{k=0}^{n} k & \sum_{k=0}^{n}{ }^{n} C_{k} k^

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Suppose, $\operatorname{det}\left[\begin{array}{cc}\sum_{k=0}^{n} k & \sum_{k=0}^{n}{ }^{n} C_{k} k^{2} \ \sum_{k=0}^{n}{ }^{n} C_{k} k & \sum_{k=0}^{n}{ }^{n} C_{k} 3^{k}\end{array}ight]=0$ holds for some positive integer $n$. Then $\sum_{k=0}^{n} \frac{{ }^{n} C_{k}}{k+1}$ equals

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(6.2) : Given, $\operatorname{det}\left[\begin{array}{cc}\sum_{k=0}^{n} k & \sum_{k=0}^{n}{ }^{n} C_{k} k^{2} \ \sum_{k=0}^{n}{ }^{n} C_{k} k & \sum_{k=0}^{n}{ }^{n} C_{k} 3^{k}\end{array}ight]=0$\ $\Rightarrow\left|\begin{array}{cc}\frac{n(n+1)}{2} & n(n+1) 2^{n-2} \ n 2^{n-1} & 4^{n}\end{array}ight|=0$\ $\Rightarrow\left|\begin{array}{cc}1 & 2^{n-1} \ n \cdot 2^{n-1} & 2^{2 n}\end{array}ight|=0 \Rightarrow\left|\begin{array}{cc}1 & 1 \ n \cdot 2^{n-1} & 2^{n+1}\end{array}ight|=0 \Rightarrow n=4$\ Now, $\sum_{k=0}^{n} \frac{{ }^{n} C_{k}}{k+1}=\sum_{k=0}^{4} \frac{{ }^{4} C_{k}}{k+1}=\frac{1}{5} \sum_{k=0}^{4}{ }^{5} C_{k+1}$\ $=\frac{1}{5}\left[2^{5}-1ight]=\frac{31}{5}=6.2$\