To find the next number in the sequence \(9, 12, 18, 105, \ ? \), let's analyze the pattern.

1. From \(9 \rightarrow 12\), there is an increase of \(12 - 9 = 3\).
2. From \(12 \rightarrow 18\), there is an increase of \(18 - 12 = 6\).
3. From \(18 \rightarrow 105\), there is a large increase of \(105 - 18 = 87\).

The differences are \(3, 6, 87\), which don't immediately form an obvious pattern.

### Possible approach:
It seems the sequence could involve multiplication or a mix of arithmetic and geometric patterns. Let's try multiplication:
- \(9 \times \frac{4}{3} = 12\)
- \(12 \times \frac{3}{2} = 18\)
- \(18 \times 5.83 \approx 105\)

Given this irregular pattern, it is hard to use standard progressions. However, if we assume a continued increase by multiplying by a growing factor, we could estimate the next value:

- If we multiply \(105\) by a factor close to \(9\) (following the rough geometric increase), we get:
  \[
  105 \times 9 = 945
  \]

Thus, the next number might be **945**.

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Do you want further details or have any questions?

Here are 5 related questions to explore:
1. What is the general formula for geometric sequences?
2. How can patterns involving both addition and multiplication be identified?
3. Can Fibonacci-like patterns apply in similar number sequences?
4. How do we predict the next number in a non-standard progression?
5. What are common tricks to spot hidden arithmetic progressions?

**Tip:** In complex sequences, look for alternating patterns of addition and multiplication, as they often combine to form non-linear sequences.

Explore how to predict the next number in the sequence 9, 12, 18, 105 by analyzing arithmetic and geometric progressions. This problem demonstrates mixed patterns with a detailed breakdown and solution. Suitable for students in Grades 8-10.

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