The given recurrence relation is $L_n = L_{n-1} + L_{n-2}$
$\therefore L_n - L_{n-1} - L_{n-2} = 0$
Therefore, the characteristic equation is $r^2 - r -1 = 0$
Solving for $r$ gives $r_1 = \frac{1+\sqrt5}{2}, r_2 = \frac{1-\sqrt5}{2}$
Therefore, the general solution is of the form $L_n = \alpha(r_1)^n + \beta(r_2)^n$
$\therefore L_n = \alpha(\frac{1+\sqrt5}{2})^n + \beta(\frac{1-\sqrt5}{2})^n$
Now, given $L_1 = 1$
$\therefore L_1 = 1 = \alpha(\frac{1+\sqrt5}{2}) + \beta(\frac{1-\sqrt5}{2}) = (\alpha + \beta) \frac{1}{2} + (\alpha - \beta) \frac{\sqrt5}{2}$
$\therefore \alpha + \beta = 2 \text{ and } \alpha - \beta = 0$
$\therefore \alpha = 1, \beta = 1$
$\therefore L_n = (\frac{1+\sqrt5}{2})^n + (\frac{1-\sqrt5}{2})^n$
Answer :- D.