Let's analyze the sequence: 2, 10, 50, ...

Checking if the sequence is arithmetic:

An arithmetic sequence has a common difference between consecutive terms.

• Difference between 10 and 2:
  $10 - 2 = 8$
• Difference between 50 and 10:
  $50 - 10 = 40$

The differences are not the same, so the sequence is not arithmetic.

Checking if the sequence is geometric:

A geometric sequence has a common ratio between consecutive terms.

• Ratio between 10 and 2:
  $\frac{10}{2} = 5$
• Ratio between 50 and 10:
  $\frac{50}{10} = 5$

The ratios are the same, so the sequence is geometric with a common ratio of $5$.

Conclusion:

The sequence is geometric with a common ratio of $5$.

Would you like further details or have any questions?

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Here are 5 related questions:

1. What would be the 5th term of this geometric sequence?
2. How do you find the sum of the first 4 terms in this sequence?
3. Can a geometric sequence have a common ratio less than 1? What would that mean?
4. How do you derive the formula for the $n$-th term of a geometric sequence?
5. What happens if the common ratio in a geometric sequence is negative?

Tip: For geometric sequences, multiplying any term by the common ratio gives the next term!