To find the next number in the sequence $2, 8, 26, 80, 242, \dots$, let's start by analyzing the pattern:

Step 1: Examine the differences between consecutive terms:

• $8 - 2 = 6$
• $26 - 8 = 18$
• $80 - 26 = 54$
• $242 - 80 = 162$

So the first-level differences are: 6, 18, 54, 162.

Step 2: Analyze the pattern of the first-level differences:

Now, observe that each difference is multiplied by 3 to get the next:

• $18 = 6 \times 3$
• $54 = 18 \times 3$
• $162 = 54 \times 3$

This suggests that the next first-level difference will be:

• $162 \times 3 = 486$

Step 3: Calculate the next term in the sequence:

To find the next term, we add this difference to the last term in the sequence:

• $242 + 486 = 728$

Conclusion:

The next number in the sequence is 728.

Would you like further details or have any questions?

Here are 5 related questions:

1. How can you identify patterns in number sequences?
2. Can you find a general formula for this sequence?
3. What other types of sequences multiply differences by a constant factor?
4. How can this pattern be applied to geometric progressions?
5. What happens when you apply higher-order differences to analyze sequences?

Tip: Always check for arithmetic, geometric, or recursive patterns when analyzing number sequences.