lean_workbook_plus_5799

Let $F_0 = 0, F_{1} = 1$ , and for $n \ge 1, F_{n+1} = F_n + F_{n-1}$ . Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$ . Then there are rational numbers $A$ and $B$ such that $\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}$ . Find $A + B$ .