Determine if the sequence below is arithmetic or geometric and determine the common difference / ratio in simplest form.
2, comma, 10, comma, 50, comma, point, point, .
2,10,50,

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!

Determine if the sequence below is arithmetic or geometric and determine the common difference / ratio in simplest form. 2, 10, 50, ...

Learn how to determine if the sequence 2, 10, 50 is arithmetic or geometric. This problem involves key concepts like finding the common difference and common ratio, suitable for students in Grades 8-10.

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