If a1 = 2 and a3 = 18 in a geometric sequence, what is the positive common ratio r?

• A. 3
• B. -3
• C. 9
• D. 1/3

Answer: (A)

In a geometric sequence each term is obtained by multiplying the previous term by a constant ratio r, and the nth term is a1 times r^(n-1). Here a1 = 2 and a3 = 18, so 18 = 2 * r^2, giving r^2 = 9. This yields r = ±3, but the ratio is required to be positive, so r = 3. Check: a2 = 2 * 3 = 6, a3 = 6 * 3 = 18, which fits. The other options don’t work with a positive ratio: 9 would make a3 far larger, and 1/3 would give a3 = 2/9, not 18. The positive common ratio is 3.