To solve the recurrence relation $T(n) = 7T(n-1) - 12T(n-2)$, we follow these steps:

Step 1: Find the Characteristic Equation

The first step is to assume that the solution is of the form $T(n) = r^n$. Substituting this into the recurrence relation gives:

$r^n = 7r^{n-1} - 12r^{n-2}$

To simplify, divide through by $r^{n-2}$:

$r^2 = 7r - 12$

This is a quadratic equation:

$r^2 - 7r + 12 = 0$

Step 2: Solve the Characteristic Equation

Solve the quadratic equation using the quadratic formula:

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $r^2 - 7r + 12 = 0$, $a = 1$, $b = -7$, and $c = 12$. Plugging in these values:

$r = \frac{7 \pm \sqrt{(-7)^2 - 4(1)(12)}}{2(1)}$ $r = \frac{7 \pm \sqrt{49 - 48}}{2}$ $r = \frac{7 \pm \sqrt{1}}{2}$

This gives us the roots:

$r = \frac{7 + 1}{2} = 4 \quad \text{and} \quad r = \frac{7 - 1}{2} = 3$

Step 3: Write the General Solution

The general solution to the recurrence relation is a linear combination of the terms corresponding to these roots:

$T(n) = A \cdot 4^n + B \cdot 3^n$

where $A$ and $B$ are constants determined by the initial conditions.

Step 4: Apply Initial Conditions (if provided)

If specific initial conditions are given, such as $T(0)$ and $T(1)$, we can solve for $A$ and $B$. If no initial conditions are provided, the solution remains in the general form:

$T(n) = A \cdot 4^n + B \cdot 3^n$

Summary

The general solution to the recurrence relation $T(n) = 7T(n-1) - 12T(n-2)$ is:

$T(n) = A \cdot 4^n + B \cdot 3^n$

where $A$ and $B$ are constants determined by the initial conditions.

Would you like to go through how to determine $A$ and $B$ if initial conditions are provided, or do you have any other questions?

Related Questions

1. How do you determine the values of $A$ and $B$ using initial conditions?
2. What if the characteristic equation has repeated roots?
3. How can you solve a non-homogeneous recurrence relation?
4. How do you solve a recurrence relation with complex roots?
5. What is the importance of the characteristic equation in solving recurrence relations?
6. Can this method be used to solve higher-order recurrence relations?
7. How would the solution differ if the characteristic equation had irrational roots?
8. What is the significance of the roots of the characteristic equation?

Tip

Always verify the roots of the characteristic equation by plugging them back into the original quadratic equation to ensure accuracy.