For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence).
75, comma, minus, 45, comma, 27, comma, point, point, .
75,−45,27,...

To determine whether the sequence \( 75, -45, 27, \ldots \) is arithmetic or geometric, we need to check for either a common difference or a common ratio.

### Step 1: Check if the sequence is arithmetic
An arithmetic sequence has a common difference, \(d\), between consecutive terms. We calculate the difference between the first and second terms and the second and third terms:

- Difference between the second term and the first term:
  \[
  -45 - 75 = -120
  \]
  
- Difference between the third term and the second term:
  \[
  27 - (-45) = 27 + 45 = 72
  \]

Since the differences are not the same (\(-120\) and \(72\)), the sequence is **not arithmetic**.

### Step 2: Check if the sequence is geometric
A geometric sequence has a common ratio, \(r\), between consecutive terms. We calculate the ratio between the second and first terms, and the third and second terms:

- Ratio between the second term and the first term:
  \[
  \frac{-45}{75} = -\frac{3}{5}
  \]

- Ratio between the third term and the second term:
  \[
  \frac{27}{-45} = -\frac{3}{5}
  \]

Since the ratios are the same (\(-\frac{3}{5}\)), the sequence is a **geometric sequence** with a common ratio of \( r = -\frac{3}{5} \).

### Conclusion:
The sequence is a geometric sequence with a common ratio of \( r = -\frac{3}{5} \).

Would you like more details or have any questions?

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Here are 5 related questions to expand this topic:

1. What is the general formula for the \(n\)-th term of a geometric sequence?
2. How would you find the 5th term of the sequence \(75, -45, 27, \dots\)?
3. Can a sequence be both arithmetic and geometric? If so, under what conditions?
4. What happens if the common ratio in a geometric sequence is greater than 1?
5. How do you determine the sum of the first \(n\) terms of a geometric sequence?

**Tip:** In a geometric sequence, the terms can alternate between positive and negative if the common ratio is negative, as seen in this sequence.

For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). 75,−45,27,...

Explore how to determine if the sequence 75, −45, 27,... is arithmetic or geometric. Learn to calculate the common difference for arithmetic sequences and the common ratio for geometric sequences. A step-by-step solution suitable for students in Grades 8-10.

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