In a geometric sequence with a1 = 3 and a3 = 27, what is the common ratio r if r is positive?

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Multiple Choice

In a geometric sequence with a1 = 3 and a3 = 27, what is the common ratio r if r is positive?

Explanation:
In a geometric sequence, each term is the first term times the common ratio raised to a power: a_n = a1 × r^(n−1). For the third term, a3 = a1 × r^2. With a1 = 3 and a3 = 27, plug in: 27 = 3 × r^2, so r^2 = 9. This gives r = ±3. Since the ratio is positive, the valid value is r = 3. Negative options would violate the positive requirement and also wouldn’t satisfy a3 = 27 when a1 = 3.

In a geometric sequence, each term is the first term times the common ratio raised to a power: a_n = a1 × r^(n−1). For the third term, a3 = a1 × r^2. With a1 = 3 and a3 = 27, plug in: 27 = 3 × r^2, so r^2 = 9. This gives r = ±3. Since the ratio is positive, the valid value is r = 3. Negative options would violate the positive requirement and also wouldn’t satisfy a3 = 27 when a1 = 3.

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