Solve the recurrence relation:
𝑎
𝑛
+
2
+
4
𝑎
𝑛
−
1
+
4
𝑎
0
=
7
 for 
𝑛
≥
0
a 
n+2
​
 +4a 
n−1
​
 +4a 
0
​
 =7 for n≥0
with the initial conditions:
𝑎
0
=
1
,
𝑎
1
=
2
a 
0
​
 =1,a 
1
​
 =2

The given recurrence relation is:

\[
a_{n+2} + 4a_{n-1} + 4a_0 = 7 \quad \text{for} \quad n \geq 0
\]

with the initial conditions:

\[
a_0 = 1, \quad a_1 = 2
\]

To solve this, let's analyze the equation step by step.

### Step 1: Plug in the initial condition \(a_0 = 1\).

\[
a_{n+2} + 4a_{n-1} + 4(1) = 7
\]
Simplifying the equation:

\[
a_{n+2} + 4a_{n-1} + 4 = 7
\]
\[
a_{n+2} + 4a_{n-1} = 3
\]

This is the simplified recurrence relation:

\[
a_{n+2} = 3 - 4a_{n-1}
\]

### Step 2: Use initial conditions to compute the first few terms.
We have \(a_0 = 1\) and \(a_1 = 2\). Let's find \(a_2\) and \(a_3\) using the recurrence relation.

For \(n = 0\):
\[
a_2 = 3 - 4a_0 = 3 - 4(1) = 3 - 4 = -1
\]

For \(n = 1\):
\[
a_3 = 3 - 4a_1 = 3 - 4(2) = 3 - 8 = -5
\]

### Step 3: Verify the pattern and calculate more terms if needed.

We now have:

\[
a_0 = 1, \quad a_1 = 2, \quad a_2 = -1, \quad a_3 = -5
\]

We could continue computing more terms using the recurrence relation if needed. Would you like to explore further steps or perhaps analyze more properties of this recurrence?

### Relative Questions:
1. What would the next term \(a_4\) be using this recurrence relation?
2. Can you express a general formula for \(a_n\)?
3. How does the recurrence relation behave for large \(n\)?
4. What would change if the initial conditions were different?
5. Can this recurrence relation be solved using generating functions?

### Tip:
When dealing with recurrence relations, solving for the first few terms helps identify patterns that might guide you towards a general solution.

Solve the recurrence relation: a_{n+2} + 4a_{n-1} + 4a_0 = 7 for n ≥ 0 with the initial conditions: a_0 = 1, a_1 = 2.

Step-by-step solution to the recurrence relation a_{n+2} + 4a_{n-1} + 4a_0 = 7 for n ≥ 0, with initial conditions a_0 = 1, a_1 = 2. Learn how to solve using recursion and calculate specific terms of the sequence.

Math Problem Statement

Solve the recurrence relation: 𝑎 𝑛 + 2 + 4 𝑎 𝑛 − 1 + 4 𝑎 0

7 for 𝑛 ≥ 0 a n+2 +4a n−1 +4a 0 =7 for n≥0 with the initial conditions: 𝑎 0

1 , 𝑎 1

Solution

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