To solve a linear homogeneous recurrence relation with given characteristic roots, the general solution is a combination of terms derived from those roots and their multiplicities.

Step-by-step process:

1. Identify the roots and their multiplicities:

   • Roots: $4, -1, 2$
   • Multiplicities: $1, 1, 3$

2. Formulate the general solution:

   • For a root $r$ with multiplicity $m$, the terms in the general solution are: $C_1 r^n, C_2 n r^n, C_3 n^2 r^n, \ldots, C_m n^{m-1} r^n$ (up to $n^{m-1} r^n$).

   • Applying this rule:

     • For $4$ (multiplicity 1): $C_1 4^n$
     • For $-1$ (multiplicity 1): $C_2 (-1)^n$
     • For $2$ (multiplicity 3): $C_3 2^n, C_4 n 2^n, C_5 n^2 2^n$

3. Combine all terms: The general solution is: $a_n = C_1 4^n + C_2 (-1)^n + C_3 2^n + C_4 n 2^n + C_5 n^2 2^n$

Final Answer:

$a_n = C_1 4^n + C_2 (-1)^n + C_3 2^n + C_4 n 2^n + C_5 n^2 2^n$

This is the general solution for the recurrence relation.

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Follow-up:

Would you like details on:

1. How to derive characteristic roots from a recurrence equation?
2. Solving recurrence relations with initial conditions?
3. An example with specific values for $a_n$?
4. Applying this method to non-homogeneous recurrence relations?
5. Understanding how multiplicities affect the solution?

Tip: Always align the general solution with the initial conditions to determine specific constants!