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.