The given recurrence relation is:

$a_n = 2a_{n-1} + 4a_{n-2}$

for $n \geq 2$. We need to find the general form of $a_n$.

Step 1: Solve the characteristic equation

This is a linear recurrence relation with constant coefficients, so we begin by solving the associated characteristic equation. The recurrence relation can be transformed into the characteristic equation by assuming a solution of the form $a_n = r^n$.

Substitute $a_n = r^n$, $a_{n-1} = r^{n-1}$, and $a_{n-2} = r^{n-2}$ into the recurrence relation:

$r^n = 2r^{n-1} + 4r^{n-2}$

Divide through by $r^{n-2}$:

$r^2 = 2r + 4$

Rearrange this to form a quadratic equation:

$r^2 - 2r - 4 = 0$

Step 2: Solve the quadratic equation

Now solve the quadratic equation $r^2 - 2r - 4 = 0$ using the quadratic formula:

$r = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-4)}}{2(1)}$

$r = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2}$

$r = \frac{2 \pm 2\sqrt{5}}{2}$

$r = 1 \pm \sqrt{5}$

So, the two roots of the characteristic equation are:

$r_1 = 1 + \sqrt{5}, \quad r_2 = 1 - \sqrt{5}$

Step 3: General solution

Since the characteristic equation has distinct real roots, the general solution to the recurrence relation is of the form:

$a_n = C_1(1 + \sqrt{5})^n + C_2(1 - \sqrt{5})^n$

where $C_1$ and $C_2$ are constants determined by the initial conditions.

If you have specific initial values for $a_0$ and $a_1$, we can proceed to find $C_1$ and $C_2$.

Would you like to provide the initial conditions, or should I continue with the general solution?

Feel free to ask if you need more details on any step. Here are some related questions:

1. What happens if the characteristic equation has complex roots?
2. How would you find the constants $C_1$ and $C_2$?
3. How can recurrence relations be solved using matrix methods?
4. Can you solve recurrence relations with variable coefficients similarly?
5. What are the applications of solving such recurrence relations in computer science?

Tip: Always check the characteristic equation first when solving linear recurrence relations with constant coefficients.