Question 1209830
You've described the Fibonacci sequence, where:

* $A_0 = 0$
* $A_1 = 1$
* $A_n = A_{n-1} + A_{n-2}$ for $n \ge 2$

We're looking for the closed-form expression of $A_n$ in the form $c \alpha^n + d \beta^n$, where $\alpha$ and $\beta$ are the roots of the characteristic equation.

**1. Find the Characteristic Equation and its Roots**

The characteristic equation for the recurrence relation is:

* $x^2 - x - 1 = 0$

Using the quadratic formula, the roots are:

* $\alpha = \frac{1 + \sqrt{5}}{2}$ (the golden ratio)
* $\beta = \frac{1 - \sqrt{5}}{2}$

**2. Use Initial Conditions to Solve for c and d**

We have the following initial conditions:

* $A_0 = 0$
* $A_1 = 1$

Substitute these into the closed-form expression:

* For $n = 0$: $A_0 = c \alpha^0 + d \beta^0 = c + d = 0$
* For $n = 1$: $A_1 = c \alpha^1 + d \beta^1 = c \left( \frac{1 + \sqrt{5}}{2} \right) + d \left( \frac{1 - \sqrt{5}}{2} \right) = 1$

From the first equation, we get $d = -c$. Substitute this into the second equation:

* $c \left( \frac{1 + \sqrt{5}}{2} \right) - c \left( \frac{1 - \sqrt{5}}{2} \right) = 1$
* $c \left( \frac{1 + \sqrt{5} - (1 - \sqrt{5})}{2} \right) = 1$
* $c \left( \frac{2 \sqrt{5}}{2} \right) = 1$
* $c \sqrt{5} = 1$
* $c = \frac{1}{\sqrt{5}}$

Now, substitute $c$ back into $d = -c$:

* $d = -\frac{1}{\sqrt{5}}$

**3. The Ordered Pair (c, d)**

Therefore, the ordered pair $(c, d)$ is:

* $\left( \frac{1}{\sqrt{5}}, -\frac{1}{\sqrt{5}} \right)$

Final Answer: The final answer is $\boxed{\left( \frac{1}{\sqrt{5}}, -\frac{1}{\sqrt{5}} \right)}$