[tex](2k+1)C_{n}^k = 2kC_{n}^k+C_{n}^k \\ \\ \\ kC_{n}^k = k\dfrac{n!}{k!(n-k)!} = \dfrac{n!}{(k-1)!(n-k)!} = \\ \\ = \dfrac{(n-1)!\,n}{(k-1)!\Big((n-1)-(k-1)\Big)!} = nC_{n-1}^{k-1}\\ \\\\ \Rightarrow (2k+1)C_{n}^k = 2nC_{n-1}^{k-1}+C_{n}^k \\ \\ S_n = C_n^0+3C_{n}^1+5C_{n}^2+...+(2n+1)C_{n}^n \\ \\ S_n = C_n^0+(2nC_{n-1}^0+C_{n}^1)+(2nC_{n-1}^1+C_{n}^2)+...+(2nC_{n-1}^{n-1}+C_{n}^n)\\ \\ S_{n} = C_{n}^0+C_{n}^1+C_{n}^2+...+C_{n}^n+2n(C_{n-1}^0+C_{n-1}^1+...+C_{n-1}^{n-1})[/tex]
[tex]S_n = 2^n+2n(2^{n-1}) \\ \\ S_n = 2^n+n2^{n} \\ \\ \Rightarrow \boxed{S_n = 2^n(n+1)}[/tex]
