Let's analyze the pattern in the given sequence:
$1, 4, 1, 12, 1, 36, 1, \dots$

Observations:

• Every second number (the even positions) increases in a certain way:
  • 4, 12, 36, ...

Pattern in the even-positioned numbers:

• From 4 to 12, the factor is 3 (since $4 \times 3 = 12$).
• From 12 to 36, the factor is also 3 (since $12 \times 3 = 36$).

Thus, the numbers in even positions follow a pattern where each number is multiplied by 3.

Predicting the next number:

• The next number in an even position (after 36) should be $36 \times 3 = 108$.

Therefore, the next number in the sequence is 108.

Final sequence:

$1, 4, 1, 12, 1, 36, 1, 108$

Would you like more details or have any questions?

Here are five related questions to deepen understanding:

1. What if the sequence only alternated between 1 and powers of 4?
2. How would the sequence change if the multiplier were 2 instead of 3?
3. Can you derive a formula for the even-position numbers?
4. How would the pattern continue if the odd-positioned numbers also followed a rule?
5. Could this sequence represent a geometric progression in some way?

Tip: Identifying patterns often requires examining different segments separately, such as odd and even positions, to uncover hidden relationships.