To analyze this sequence, let's try to identify the pattern or rule that defines it.

The sequence is: $9, 12, 18, 105, \_$

Step 1: Investigate the differences

Let's check the differences between consecutive terms:

• $12 - 9 = 3$
• $18 - 12 = 6$
• $105 - 18 = 87$

The differences don't follow a simple arithmetic progression, but we can see that they increase in magnitude.

Step 2: Check for ratios (geometric sequence)

Now, let’s check if there is a ratio between the terms:

• $\frac{12}{9} = \frac{4}{3}$
• $\frac{18}{12} = \frac{3}{2}$
• $\frac{105}{18} \approx 5.833$

The ratios are also not constant, so it isn't a geometric sequence either.

Step 3: Examine potential factorial or mixed patterns

Given that both the differences and ratios vary, the pattern might be a combination of arithmetic or geometric growth followed by a sharp jump in value. The large increase from 18 to 105 suggests a possible switch in rules or an exceptional case.

Step 4: Hypothesis and next term

One possible hypothesis is that the next term might follow a distinct multiplication rule. Given that the previous large jump was approximately by a factor of 5.83, applying a similar factor to the next term:

• $105 \times 6 \approx 630$

So, the next number in the sequence is likely 630.

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Would you like further details or have any questions about this analysis?

Here are 5 related questions for deeper exploration:

1. Can the sequence be modeled with a recursive formula?
2. What if the sequence were geometric instead of mixed?
3. How does the jump from 18 to 105 fit into common sequence types?
4. What other types of sequences display non-linear jumps like this?
5. Can a polynomial function be derived to fit this sequence?

Tip: Sometimes sequences have mixed patterns where certain jumps follow arithmetic or geometric rules, while others introduce a sharp change. Always try both approaches.