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.