Solve the recurrence relationa_r=-2a_(r-1)-a_(r-2)+5

To solve the recurrence relation:

\[
a_r = -2a_{r-1} - a_{r-2} + 5,
\]

we can break it into two parts: solving the homogeneous part and finding a particular solution.

### 1. Homogeneous Solution

Consider the homogeneous part of the recurrence relation:

\[
a_r = -2a_{r-1} - a_{r-2}.
\]

The characteristic equation associated with this is:

\[
x^2 + 2x + 1 = 0.
\]

This factors as:

\[
(x + 1)^2 = 0.
\]

Thus, the solution to the homogeneous equation is:

\[
a_r^{(h)} = (C_1 + C_2r)(-1)^r,
\]
where \(C_1\) and \(C_2\) are constants to be determined by initial conditions.

### 2. Particular Solution

Now, let's find a particular solution to the non-homogeneous recurrence:

\[
a_r = -2a_{r-1} - a_{r-2} + 5.
\]

Since the non-homogeneous part is a constant (5), we can try a particular solution of the form \(a_r^{(p)} = A\), where \(A\) is a constant. Substituting this into the original recurrence gives:

\[
A = -2A - A + 5.
\]

Simplifying this:

\[
A = -3A + 5 \implies 4A = 5 \implies A = \frac{5}{4}.
\]

Thus, the particular solution is \(a_r^{(p)} = \frac{5}{4}\).

### 3. General Solution

The general solution to the recurrence relation is the sum of the homogeneous and particular solutions:

\[
a_r = (C_1 + C_2r)(-1)^r + \frac{5}{4}.
\]

### 4. Initial Conditions

If initial conditions are provided, we can determine \(C_1\) and \(C_2\). Without them, this is the most general form of the solution.

---

Do you want to proceed with applying specific initial conditions or ask for any clarifications?

Here are 5 related questions to explore further:
1. How do you solve recurrence relations with complex roots in the characteristic equation?
2. What method is used to solve recurrence relations with non-constant coefficients?
3. How would the solution change if the non-homogeneous term were a function of \(r\) instead of a constant?
4. How do recurrence relations connect to difference equations in discrete mathematics?
5. Can you explain the role of generating functions in solving recurrence relations?

**Tip:** Recurrence relations are closely related to linear difference equations. Understanding the characteristic equation is key to solving them efficiently.

Solve the recurrence relation a_r = -2a_(r-1) - a_(r-2) + 5.

Explore how to solve the recurrence relation a_r = -2a_(r-1) - a_(r-2) + 5 with a detailed solution. Learn the steps to solve the homogeneous part, find a particular solution, and derive the general solution. Suitable for college students or advanced high school learners.

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