To determine whether the sequence $2, 8, 26, 80, \dots$ is arithmetic, geometric, or neither, we need to check if there is a common difference (for arithmetic) or a common ratio (for geometric).

Checking for an Arithmetic Sequence:

An arithmetic sequence has a constant difference between consecutive terms. We calculate the differences between consecutive terms:

1. $8 - 2 = 6$
2. $26 - 8 = 18$
3. $80 - 26 = 54$

Since the differences are not the same ($6, 18, 54$), this is not an arithmetic sequence.

Checking for a Geometric Sequence:

A geometric sequence has a constant ratio between consecutive terms. We calculate the ratios between consecutive terms:

1. $\frac{8}{2} = 4$
2. $\frac{26}{8} = 3.25$
3. $\frac{80}{26} \approx 3.08$

Since the ratios are not the same ($4, 3.25, 3.08$), this is not a geometric sequence.

Conclusion:

Since there is no common difference or common ratio, the sequence is neither arithmetic nor geometric.

Thus, the correct answer is:

Neither; There is no common ratio and no common difference.

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Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:

1. What is the formula for the $n$-th term of an arithmetic sequence?
2. How do you find the common ratio of a geometric sequence?
3. Can a sequence be both arithmetic and geometric?
4. What is the difference between an arithmetic progression and an arithmetic series?
5. How do you determine if a sequence is quadratic?

Tip: When working with sequences, always try to check both for common differences and common ratios to classify them correctly.