利用\(kC_{k}^{n}=nC_{k-1}^{n-1}\)
\(\begin{align}
  & {{\left( C_{1}^{n} \right)}^{2}}+2{{\left( C_{2}^{n} \right)}^{2}}+3{{\left( C_{3}^{n} \right)}^{2}}+\cdots +n{{\left( C_{n}^{n} \right)}^{2}} \\
& =C_{1}^{n}C_{n-1}^{n}+2C_{2}^{n}C_{n-2}^{n}+3C_{3}^{n}C_{n-3}^{n}+\cdots +nC_{n}^{n}C_{0}^{n} \\
& =nC_{0}^{n-1}C_{n-1}^{n}+nC_{1}^{n-1}C_{n-2}^{n}+nC_{2}^{n-1}C_{n-3}^{n}+\cdots +nC_{n-1}^{n-1}C_{0}^{n} \\
& =...... \\
\end{align}\)

應該是這樣
\(\begin{align}
  & {{x}_{n+1}}+{{y}_{n+1}}\sqrt{3}={{\left( 2+\sqrt{3} \right)}^{n+1}}=\left( {{x}_{n}}+{{y}_{n}}\sqrt{3} \right)\left( 2+\sqrt{3} \right)=\left[ \left( 2{{x}_{n}}+3{{y}_{n}} \right)+\left( {{x}_{n}}+2{{y}_{n}} \right)\sqrt{3} \right] \\
& \frac{{{x}_{n+1}}}{{{y}_{n+1}}}=\frac{2{{x}_{n}}+3{{y}_{n}}}{{{x}_{n}}+2{{y}_{n}}}=\frac{2\left( \frac{{{x}_{n}}}{{{y}_{n}}} \right)+3}{\left( \frac{{{x}_{n}}}{{{y}_{n}}} \right)+2} \\
\end{align}\)