Let $$\alpha=\sum_\limits{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)$$ and $$\beta=\sum_\limits{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)$$ If $$5 \alpha=6 \beta$$, then $$n$$ equals _______.

Let $$S_n$$ be the sum to $$n$$-terms of an arithmetic progression $$3,7,11$$, If $$40<\left(\frac{6}{n(n+1)} \sum_\limits{k=1}^n S_k\right)<42$$, then $$n$$ equals ________.

If 50 Vernier divisions are equal to 49 main scale divisions of a traveling microscope and one smallest reading of main scale is $$0.5 \mathrm{~mm}$$, the Vernier constant of traveling microscope is

An electron revolving in $$n^{\text {th }}$$ Bohr orbit has magnetic moment $$\mu_n$$. If $$\mu_n \propto n^x$$, the value of $$x$$ is