what are the next 3 in the series
12,20,21,27,100

To identify the next three numbers in the sequence \(12, 20, 21, 27, 100\), let's analyze the pattern:

1. **From 12 to 20:** The difference is \(20 - 12 = 8\).
2. **From 20 to 21:** The difference is \(21 - 20 = 1\).
3. **From 21 to 27:** The difference is \(27 - 21 = 6\).
4. **From 27 to 100:** The difference is \(100 - 27 = 73\).

These differences \(8, 1, 6, 73\) don't immediately suggest a simple arithmetic or geometric progression. However, let's hypothesize that the pattern might involve varying differences or an underlying formula, but given the sequence’s irregularity, it could also be non-mathematical or based on an external rule.

Since there’s no clear numerical pattern, the next three numbers could be speculative without additional context. However, we could consider the possibility of increasing large gaps:

- After a jump of \(73\) (from \(27\) to \(100\)), the next jump could be similarly large or unpredictable. 

If we hypothesize larger increases, possible next numbers could be:

1. **Next Number (based on similar increasing large difference):** \(100 + (73 + x)\), where \(x\) is some increment. A speculative next number might be \(100 + 100 = 200\) (for instance).
2. **Continuing this pattern:** Another significant jump could be \(200 + 150 = 350\).
3. **Another large increment:** Then, perhaps \(350 + 250 = 600\).

So, one possible set of next numbers could be:

**Next three numbers:** 200, 350, 600.

Would you like details on this or have any specific questions about the reasoning? 

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**Related questions:**

1. What if the differences were based on a non-linear pattern?
2. Could a logarithmic or factorial pattern apply to this sequence?
3. How do sequences like these appear in real-world contexts?
4. What are examples of irregular sequences in mathematical puzzles?
5. Could external factors or constraints define this sequence?

**Tip:** When identifying patterns in number sequences, consider not only arithmetic and geometric progressions but also more complex operations like factorials, exponentials, or even non-mathematical rules.

Explore the sequence 12, 20, 21, 27, 100 and identify its pattern. This problem involves analyzing number sequences, specifically focusing on arithmetic progression and irregular sequences. Suitable for students in Grades 7-9.

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