3. The general solution of the recursive relation: 𝑎𝑛 = 𝑎𝑛-1 +6𝑎𝑛-2, is... 

To find the general solution of the recursive relation \( a_n = a_{n-1} + 6a_{n-2} \), we follow these steps:

1. **Find the characteristic equation**:
   The given recurrence relation is a linear homogeneous recurrence relation with constant coefficients. We can write the characteristic equation by assuming \( a_n = r^n \) as a solution:
   \[
   r^n = r^{n-1} + 6r^{n-2}
   \]
   Dividing both sides by \( r^{n-2} \) (assuming \( r \neq 0 \)):
   \[
   r^2 = r + 6
   \]
   This simplifies to the characteristic equation:
   \[
   r^2 - r - 6 = 0
   \]

2. **Solve the characteristic equation**:
   To find the roots of the characteristic equation \( r^2 - r - 6 = 0 \), we use the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -1 \), and \( c = -6 \):
   \[
   r = \frac{1 \pm \sqrt{1 + 24}}{2}
   \]
   \[
   r = \frac{1 \pm 5}{2}
   \]
   Therefore, the roots are:
   \[
   r_1 = \frac{1 + 5}{2} = 3 \quad \text{and} \quad r_2 = \frac{1 - 5}{2} = -2
   \]

3. **Form the general solution**:
   Since the roots of the characteristic equation are distinct, the general solution to the recurrence relation is a linear combination of the terms \( r_1^n \) and \( r_2^n \):
   \[
   a_n = A \cdot 3^n + B \cdot (-2)^n
   \]
   where \( A \) and \( B \) are constants determined by the initial conditions.

Hence, the general solution of the recurrence relation \( a_n = a_{n-1} + 6a_{n-2} \) is:
\[
a_n = A \cdot 3^n + B \cdot (-2)^n
\]

Would you like more details on any part of this process or have any questions?

Here are 5 related questions you might be interested in:
1. How do you determine the initial conditions for a recurrence relation?
2. What happens if the characteristic equation has repeated roots?
3. Can you explain the method for solving non-homogeneous recurrence relations?
4. How do you convert a non-homogeneous recurrence relation into a homogeneous one?
5. What is the significance of the characteristic equation in solving recurrence relations?

**Tip:** Always verify the roots of the characteristic equation as they form the foundation for finding the general solution of the recurrence relation.

Explore the general solution of the recursive relation a_n = a_(n-1) + 6a_(n-2) using the characteristic equation method. Understand how to derive the solution step-by-step, including finding roots and forming the general formula. Suitable for undergraduate students studying recurrence relations and linear algebra.

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