To find the next number in the sequence, we should first analyze the pattern:

The sequence is: 12, 20, 24, 28, 34, 46, 76, ?

1. 12 to 20: +8
2. 20 to 24: +4
3. 24 to 28: +4
4. 28 to 34: +6
5. 34 to 46: +12
6. 46 to 76: +30

To predict the next number, we observe that the differences between consecutive terms aren't consistent. However, let's analyze the pattern in the differences:

• 8, 4, 4, 6, 12, 30

The pattern of differences increases irregularly, but we can hypothesize that the pattern involves some multiplication or addition by progressively larger values.

If we look closely:

• From 8 to 4, it's a decrease.
• From 4 to 6, it's an increase by 2.
• From 6 to 12, it's an increase by 6.
• From 12 to 30, it's an increase by 18.

Notice, 2, 6, 18 are multiplied by 3, indicating that the next difference might be 30 * 3 = 90.

So, if we add 90 to the last number in the sequence:

$76 + 90 = 166$

Thus, the next number in the sequence is likely 166.

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Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the sequence change if the differences followed a different pattern?
2. What are arithmetic and geometric sequences, and how do they differ from this sequence?
3. Can we find a formula to predict any term in this sequence?
4. What other types of sequences involve increasing or decreasing differences?
5. How can we use sequences in real-life applications?

Tip: When analyzing a sequence, always check the differences between terms first. If the differences aren't constant, consider the differences of the differences, as this might reveal a hidden pattern.