There seems to be a minor issue with the problem you typed. I’ll assume you meant the following recurrence relation:

$a_n = 2a_{n-1} + 3, \quad \text{with} \quad a_0 = 6.$

Let's solve this step-by-step.

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1. Calculate Initial Terms:

Using the recurrence relation $a_n = 2a_{n-1} + 3$:

• $a_0 = 6$
• $a_1 = 2a_0 + 3 = 2(6) + 3 = 12 + 3 = 15$
• $a_2 = 2a_1 + 3 = 2(15) + 3 = 30 + 3 = 33$
• $a_3 = 2a_2 + 3 = 2(33) + 3 = 66 + 3 = 69$

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2. General Formula:

This is a non-homogeneous linear recurrence relation. We can solve it using the method of characteristic equations and a particular solution.

Step 1: Solve the Homogeneous Equation

The homogeneous version is:

$a_n^{(h)} = 2a_{n-1}.$

The solution to this is:

$a_n^{(h)} = C \cdot 2^n, \quad \text{where} \quad C \text{ is a constant.}$

Step 2: Find a Particular Solution

For the non-homogeneous part $2a_{n-1} + 3$, we try a constant particular solution $a_n^{(p)} = A$. Substituting into the recurrence relation:

$A = 2A + 3.$

Rearranging:

$- A = 3 \quad \Rightarrow \quad A = -3.$

So, the particular solution is $a_n^{(p)} = -3$.

Step 3: General Solution

The general solution is:

$a_n = a_n^{(h)} + a_n^{(p)} = C \cdot 2^n - 3.$

Step 4: Find $C$

Using the initial condition $a_0 = 6$:

$6 = C \cdot 2^0 - 3 \quad \Rightarrow \quad 6 = C - 3 \quad \Rightarrow \quad C = 9.$

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3. Final Solution

The closed-form solution is:

$a_n = 9 \cdot 2^n - 3.$

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4. Verify the Formula (Optional)

Let's verify with the initial terms:

• $n = 0$: $a_0 = 9 \cdot 2^0 - 3 = 9 - 3 = 6$.
• $n = 1$: $a_1 = 9 \cdot 2^1 - 3 = 18 - 3 = 15$.
• $n = 2$: $a_2 = 9 \cdot 2^2 - 3 = 36 - 3 = 33$.
• $n = 3$: $a_3 = 9 \cdot 2^3 - 3 = 72 - 3 = 69$.

The formula works!

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Do you have any further questions, or would you like to explore this solution in more detail?

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5 Follow-Up Questions:

1. How do we derive solutions for more complex recurrence relations?
2. What happens if the initial condition is different, say $a_0 = 10$?
3. Can we use generating functions to solve this recurrence relation?
4. What is the behavior of $a_n$ as $n \to \infty$?
5. How does the solution change if the recurrence relation is $a_n = 3a_{n-1} + 3$?

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Tip: Always check if a recurrence relation is linear and homogeneous; it simplifies the solution method significantly.