What is the next number in the given sequence: 2, 8, 26, 80, 242, _?

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.

Discover how to find the next number in the sequence 2, 8, 26, 80, 242 using a step-by-step approach. Learn how to recognize patterns in number sequences by analyzing first-level differences and applying geometric progression principles. Suitable for high school students in Grades 9-12.

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