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.