T(n) = 7T(n-1) - 12T(n-2), solve using recurrence relation method

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.

Learn how to solve the recurrence relation T(n) = 7T(n-1) - 12T(n-2) using the characteristic equation method. This problem involves advanced concepts in mathematics, including recurrence relations and quadratic equations. Suitable for advanced mathematics students and professionals.

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