To solve the recurrence relation

$a_n = -3a_{n-1} - 3a_{n-2} - a_{n-3} \quad \text{for } n \geq 2$

with the initial conditions $a_0 = 5$, $a_1 = -9$, and $a_2 = 15$, we proceed as follows:

Step 1: Set up the characteristic equation

The associated characteristic equation for the recurrence relation is found by assuming a solution of the form $a_n = r^n$. Substituting this into the recurrence relation gives:

$r^n = -3r^{n-1} - 3r^{n-2} - r^{n-3}$

Dividing through by $r^{n-3}$ (assuming $r \neq 0$):

$r^3 + 3r^2 + 3r + 1 = 0$

Step 2: Solve the characteristic equation

The characteristic equation can be factored as follows:

$r^3 + 3r^2 + 3r + 1 = (r + 1)^3 = 0$

So, the characteristic equation has a repeated root $r = -1$ with multiplicity 3.

Step 3: Write the general solution

Since $r = -1$ is a repeated root with multiplicity 3, the general solution to the recurrence relation is:

$a_n = (C_1 + C_2n + C_3n^2)(-1)^n$

where $C_1$, $C_2$, and $C_3$ are constants to be determined from the initial conditions.

Step 4: Apply the initial conditions

Now, we use the initial conditions to determine $C_1$, $C_2$, and $C_3$.

1. For $n = 0$: $a_0 = C_1(-1)^0 = C_1 = 5$ So, $C_1 = 5$.

2. For $n = 1$: $a_1 = (C_1 + C_2 \cdot 1 + C_3 \cdot 1^2)(-1)^1 = -(C_1 + C_2 + C_3)$ Given $a_1 = -9$: $-(5 + C_2 + C_3) = -9 \implies 5 + C_2 + C_3 = 9 \implies C_2 + C_3 = 4$

3. For $n = 2$: $a_2 = (C_1 + C_2 \cdot 2 + C_3 \cdot 2^2)(-1)^2 = C_1 + 2C_2 + 4C_3$ Given $a_2 = 15$: $5 + 2C_2 + 4C_3 = 15 \implies 2C_2 + 4C_3 = 10 \implies C_2 + 2C_3 = 5$

Step 5: Solve for $C_2$ and $C_3$

We now have the system of linear equations: $C_2 + C_3 = 4$ $C_2 + 2C_3 = 5$

Subtract the first equation from the second: $(C_2 + 2C_3) - (C_2 + C_3) = 5 - 4 \implies C_3 = 1$

Substitute $C_3 = 1$ into the first equation: $C_2 + 1 = 4 \implies C_2 = 3$

Final Answer

The general solution is:

$a_n = (5 + 3n + n^2)(-1)^n$

This is the closed-form expression for the sequence.

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Related Questions

1. How would you solve a recurrence relation with complex roots in the characteristic equation?
2. What is the impact of different initial conditions on the solution to a recurrence relation?
3. How does the multiplicity of roots affect the general solution of a recurrence relation?
4. Can the method used here be extended to non-homogeneous recurrence relations?
5. How would you approach solving a non-linear recurrence relation?

Tip

When solving recurrence relations, always check the multiplicity of the roots of the characteristic equation, as it significantly affects the structure of the general solution.