https://file.aiquickdraw.com/mathbot/file/1725874310857g92zjzji.jpg

The recurrence relation provided is:

\[
a_n = a_{n-1} + 2a_{n-2}, \quad \text{with } a_0 = 2 \text{ and } a_1 = 7.
\]

### To solve this recurrence relation:
1. **Find the characteristic equation**:  
   The homogeneous recurrence relation is \( a_n = a_{n-1} + 2a_{n-2} \), which can be transformed into the characteristic equation:
   \[
   r^n = r^{n-1} + 2r^{n-2}
   \]
   Dividing both sides by \( r^{n-2} \), we get:
   \[
   r^2 = r + 2
   \]
   Rearranging terms:
   \[
   r^2 - r - 2 = 0
   \]
   This quadratic equation can be factored as:
   \[
   (r - 2)(r + 1) = 0
   \]
   Hence, the solutions to the characteristic equation are \( r_1 = 2 \) and \( r_2 = -1 \).

2. **General solution**:  
   The general solution to the recurrence relation is of the form:
   \[
   a_n = c_1 \cdot 2^n + c_2 \cdot (-1)^n
   \]
   where \( c_1 \) and \( c_2 \) are constants to be determined from the initial conditions.

3. **Use the initial conditions** to solve for \( c_1 \) and \( c_2 \):
   - From \( a_0 = 2 \):
     \[
     a_0 = c_1 \cdot 2^0 + c_2 \cdot (-1)^0 = c_1 + c_2 = 2
     \]
   - From \( a_1 = 7 \):
     \[
     a_1 = c_1 \cdot 2^1 + c_2 \cdot (-1)^1 = 2c_1 - c_2 = 7
     \]

4. **Solve the system of equations**:
   \[
   \begin{cases}
   c_1 + c_2 = 2 \\
   2c_1 - c_2 = 7
   \end{cases}
   \]
   Add both equations:
   \[
   3c_1 = 9 \quad \Rightarrow \quad c_1 = 3
   \]
   Substitute \( c_1 = 3 \) into the first equation:
   \[
   3 + c_2 = 2 \quad \Rightarrow \quad c_2 = -1
   \]

5. **Final solution**:
   The solution to the recurrence relation is:
   \[
   a_n = 3 \cdot 2^n - (-1)^n
   \]

Would you like to see further details or clarifications?

---

### Related questions:
1. What are characteristic equations, and how do they help in solving recurrence relations?
2. Can this method be applied to non-homogeneous recurrence relations?
3. How does changing the initial conditions affect the solution of a recurrence relation?
4. What is the difference between linear and non-linear recurrence relations?
5. How do you solve recurrence relations with repeated roots in their characteristic equations?

### Tip:
When solving recurrence relations, always verify that your characteristic equation has real roots before proceeding with the general solution.

What is the solution of the recurrence relation a_n = a_{n-1} + 2a_{n-2} with a_0 = 2 and a_1 = 7?

Learn how to solve the recurrence relation a_n = a_{n-1} + 2a_{n-2} with initial conditions a_0 = 2 and a_1 = 7. This guide covers characteristic equations and provides a step-by-step solution using algebraic methods. Suitable for university-level students studying discrete mathematics or algorithms.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation