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.