Question

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

• A. \(L_n = \left(\tfrac{1+\sqrt{5}}{2}\right)^n + \left(\tfrac{1-\sqrt{5}}{2}\right)^n\)
• B. \(L_n = \left(\tfrac{1+\sqrt{5}}{2}\right)^n - \left(\tfrac{1-\sqrt{5}}{3}\right)^n\)
• C. \(L_n = \left(\tfrac{1+\sqrt{5}}{2}\right)^n + \left(\tfrac{1-\sqrt{5}}{3}\right)^n\)
• D. \(L_n = \left(\tfrac{1+\sqrt{5}}{2}\right)^n - \left(\tfrac{1-\sqrt{5}}{2}\right)^n\)

Hint:

The Lucas sequence is closely related to the Fibonacci sequence. While Fibonacci numbers use $\frac{\alpha^n - \beta^n}{\alpha - \beta}$, the Lucas numbers use $\alpha^n + \beta^n$.

Answer 1

The Correct Option is A

To determine the correct expression for the Lucas sequence \(L_n\), let's examine the recurrence relation and given options:

The Lucas sequence is defined as: \[ L_n = L_{n-1} + L_{n-2} \, \text{for } n \geq 3 \] with initial conditions: \(L_1 = 1\) and \(L_2 = 3\).

This is similar to the Fibonacci sequence but starts with different initial values. The Lucas sequence has a closed-form expression involving the golden ratio \(\phi\) and its conjugate.

Recall that the closed-form expressions for such sequences involve the roots of the characteristic equation. The characteristic equation for the recurrence relation above is: \[ x^2 = x + 1 \Rightarrow x^2 - x - 1 = 0 \]

Solving this quadratic equation gives two roots: \[ \phi = \frac{1+\sqrt{5}}{2} \quad \text{and} \quad \psi = \frac{1-\sqrt{5}}{2} \]

These are the same roots used in the closed-form solution for the Fibonacci sequence. The closed-form expression for the Lucas sequence is: \[ L_n = \phi^n + \psi^n \]

Now, let's check the given options:

1. \( L_n = \left(\frac{1+\sqrt{5}}{2}\right)^n + \left(\frac{1-\sqrt{5}}{2}\right)^n \)
   This matches the derived formula for the Lucas sequence. Therefore, this option is correct.
2. \( L_n = \left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{3}\right)^n \)
   This option is incorrect because the second term has an incorrect denominator.
3. \( L_n = \left(\frac{1+\sqrt{5}}{2}\right)^n + \left(\frac{1-\sqrt{5}}{3}\right)^n \)
   This option is incorrect as the exponent's base in the second term is incorrect.
4. \( L_n = \left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n \)
   This option is incorrect as well because it subtracts rather than adding the terms.

The correct option is: \( L_n = \left(\tfrac{1+\sqrt{5}}{2}\right)^n + \left(\tfrac{1-\sqrt{5}}{2}\right)^n \)