To solve this recurrence relation for the sequence \(\{a_n\}\), we have:

1. **Initial condition**:
   \[
   a_1 = 0
   \]

2. **Recursive formula**:
   \[
   a_n = 1 + a_{n-1} + 2n - 2
   \]

This means each term is defined in terms of the previous term with an additional value that includes both a constant and a term that depends on \(n\).

Let’s calculate the first few terms to see if we can identify a pattern:

- For \( n = 1 \):
  \[
  a_1 = 0 \quad \text{(initial condition)}
  \]

- For \( n = 2 \):
  \[
  a_2 = 1 + a_1 + 2 \cdot 2 - 2 = 1 + 0 + 4 - 2 = 3
  \]

- For \( n = 3 \):
  \[
  a_3 = 1 + a_2 + 2 \cdot 3 - 2 = 1 + 3 + 6 - 2 = 8
  \]

- For \( n = 4 \):
  \[
  a_4 = 1 + a_3 + 2 \cdot 4 - 2 = 1 + 8 + 8 - 2 = 15
  \]

It appears the terms are growing quadratically. Solving the recurrence more formally would involve finding a closed-form expression for \(a_n\), but from these calculations, we see that each term builds on the previous by adding a quantity that increases linearly with \(n\).

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Would you like further analysis on finding a closed-form expression or details on the derivation of a solution? 

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**Questions to explore:**
1. How does one solve a recurrence relation using characteristic equations?
2. Can we apply the method of iteration to guess a pattern for \(a_n\)?
3. What is the relationship between recurrence relations and sequences in general?
4. How might we confirm the closed-form formula for a recurrence sequence?
5. How does changing the initial condition affect the solution?

**Tip:** When working with recurrence relations, computing the first few terms often helps in identifying patterns, which is a good step towards finding a solution.

Explore the process of solving the recurrence relation an = 1 + an-1 + 2n - 2 with an initial condition of a1 = 0. This step-by-step solution covers the iterative approach and discusses how the sequence terms grow. Ideal for students in Grades 11-12.

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