https://file.aiquickdraw.com/mathbot/file/1726533525404jz9w3t4t.jpg

The image you have uploaded contains a sequence of numbers and a recurrence relation. Let's break down what is present:

1. **Number sequence**: `0, 2, 4, 6, 8, _ _` - It appears to be an arithmetic progression where the difference between each term is `+2`.

2. **Recurrence relation**: You’ve written something that resembles a recurrence relation, specifically:
   \[
   a_n = a_{n-1} \cdot (n)
   \]
   This means that each term is determined by multiplying the previous term by \( n \) (where \( n \) could represent the position in the sequence).

### Completing the sequence:
The sequence `0, 2, 4, 6, 8` increases by 2 each time. Therefore, to fill in the blanks:
- After 8, the next number would be **10** (since \(8 + 2 = 10\)).
- After 10, the next number would be **12** (since \(10 + 2 = 12\)).

### Recap:
The completed sequence is:
\[
0, 2, 4, 6, 8, 10, 12
\]

Would you like more details on how arithmetic sequences or recurrence relations work?

Here are 5 questions you could explore next:
1. How do you formally define an arithmetic progression?
2. What is a general formula for the \(n\)-th term of an arithmetic sequence?
3. How does a recurrence relation work in generating sequences?
4. Can this pattern be modeled with a different type of sequence, like geometric progression?
5. How can you solve for unknown terms if the common difference is not given?

**Tip**: Arithmetic sequences can always be identified by a constant difference between consecutive terms.

What is the next number in the sequence 0, 2, 4, 6, 8, _ _ and how is the recurrence relation defined?

Discover how to complete the arithmetic sequence 0, 2, 4, 6, 8, and understand how recurrence relations work. This page covers arithmetic progressions, recurrence formulas, and step-by-step explanations suitable for students in Grades 6-8.

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