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The question asks: 

**"What is the general solution of a linear homogeneous recurrence relation if its characteristic equation has the roots \( r = 1, -1, -1 \)?"**

### Step-by-Step Solution:

1. **Linear Homogeneous Recurrence Relation:**
   The general form of such a recurrence relation is:
   \[
   a_n = c_1 a_{n-1} + c_2 a_{n-2} + \cdots + c_k a_{n-k}
   \]
   where \( a_n \) is the sequence, and \( c_1, c_2, \dots, c_k \) are constants. To find the general solution, we solve the **characteristic equation** of the form:
   \[
   r^k + c_1 r^{k-1} + \cdots + c_k = 0
   \]
   
2. **Given Characteristic Roots:**
   The roots of the characteristic equation are provided as:
   \[
   r_1 = 1, r_2 = -1, r_3 = -1
   \]
   Notice that the root \( -1 \) has **multiplicity 2** (i.e., it appears twice).

3. **General Solution Formula:**
   The general solution depends on the roots of the characteristic equation:
   - If a root \( r \) has multiplicity 1, the corresponding term in the solution is \( A r^n \).
   - If a root \( r \) has multiplicity 2, the corresponding terms in the solution are \( B r^n \) and \( C n r^n \).

   For the given roots:
   - For \( r_1 = 1 \), the term is \( A \cdot 1^n = A \).
   - For \( r_2 = -1 \) (with multiplicity 2), the terms are \( B (-1)^n \) and \( C n (-1)^n \).

4. **General Solution:**
   Therefore, the general solution to the recurrence relation is:
   \[
   a_n = A + B (-1)^n + C n (-1)^n
   \]
   where \( A \), \( B \), and \( C \) are constants determined by initial conditions.

Would you like more details on solving such recurrence relations? 

Here are some follow-up questions:
1. What are the initial conditions needed to fully determine the solution?
2. How does the multiplicity of a root affect the general solution?
3. Can this method be extended to non-homogeneous recurrence relations?
4. What happens if all roots are distinct?
5. What is the process of deriving the characteristic equation from the recurrence relation?

**Tip:** Always check the multiplicity of each root to ensure you're adding the correct number of terms to the general solution.

What is the general solution of a linear homogeneous recurrence relation if its characteristic equation has the roots r = 1, -1, -1?

Learn how to solve a linear homogeneous recurrence relation with the characteristic equation roots r = 1, -1, -1. This guide explains the step-by-step solution for repeated roots and provides the general formula. Suitable for undergraduate students studying algebra and recurrence relations.

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