determine if the sequence is arithmetic or geometric and determine the common difference / ratio in simplest form 15,10,5

Let's determine whether the sequence \( 15, 10, 5 \) is arithmetic or geometric.

### 1. Check if the sequence is **arithmetic**:
- In an arithmetic sequence, the difference between consecutive terms is constant.
- Subtract the second term from the first:
  \[
  10 - 15 = -5
  \]
- Subtract the third term from the second:
  \[
  5 - 10 = -5
  \]
- Since the difference is the same, the sequence is **arithmetic**.

- The common difference (\(d\)) is \( -5 \).

### 2. Check if the sequence is **geometric**:
- In a geometric sequence, the ratio between consecutive terms is constant.
- Divide the second term by the first:
  \[
  \frac{10}{15} = \frac{2}{3}
  \]
- Divide the third term by the second:
  \[
  \frac{5}{10} = \frac{1}{2}
  \]
- Since the ratios are different, the sequence is **not geometric**.

### Conclusion:
- The sequence \( 15, 10, 5 \) is **arithmetic**, with a common difference of \( -5 \).

Would you like more details on this, or do you have any questions?

Here are some related questions you might want to explore:
1. What are the formulas for the nth term of an arithmetic sequence?
2. How can you determine the sum of the first n terms of an arithmetic sequence?
3. How would this sequence look if it were geometric instead?
4. Can a sequence be both arithmetic and geometric?
5. What is the graphical representation of an arithmetic sequence?

**Tip:** For arithmetic sequences, the nth term can be calculated using the formula \( a_n = a_1 + (n-1) \cdot d \).

Learn how to determine if the sequence 15, 10, 5 is arithmetic or geometric, and find the common difference. This problem covers key concepts in arithmetic and geometric sequences, suitable for students in Grades 6-8.

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