The Lucas sequence $L_n$ is defined by the recurrence relation: $$L_n = L_{n-1} + L_{n-2}, \quad \text{for } n \ge 3,$$ with $L_1 = 1$ and $L_2 = 3$. Which one of the options given is TRUE?

The recurrence relation is given by:

The characteristic equation is:

Solving for the roots $r$ using the quadratic formula:

Let $\alpha = \frac{1+\sqrt{5}}{2}$ and $\beta = \frac{1-\sqrt{5}}{2}$. The general solution to the recurrence relation is:

We can find the constants $c_1$ and $c_2$ using the initial conditions $L_1 = 1$ and $L_2 = 3$:

1. For $n = 1$:

2. For $n = 2$:

Since $\alpha^2 = \frac{3+\sqrt{5}}{2}$ and $\beta^2 = \frac{3-\sqrt{5}}{2}$:

Subtracting the first equation from the second equation:

Substituting $c_1 + c_2 = 2$ back into the first equation:

Since $c_1 + c_2 = 2$ and $c_1 = c_2$, we obtain $c_1 = 1$ and $c_2 = 1$.

Therefore, the closed-form expression is:

This matches option A.