Let α and β be the roots of x^2+√ 3 x-16=0, and γ and δ be the roots of x^2+3 x-1=0. If P_n= α^n+β^n and Q_n=γ^n+δ^n, then (P_25+√ 3 P_24)/(2…

Let $\displaystyle \alpha$ and $\displaystyle \beta$ be the roots of $\displaystyle x^2+\sqrt{3} x-16=0$, and $\displaystyle \gamma$ and $\displaystyle \delta$ be the roots of $\displaystyle x^2+3 x-1=0$. If $\displaystyle P_n=$ $\displaystyle \alpha^n+\beta^n$ and $\displaystyle Q_n=\gamma^n+\delta^n$, then $\displaystyle \frac{P_{25}+\sqrt{3} P_{24}}{2 P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to