For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). 12,quad8sqrt3,quad16,dots 2sqrt3 (sqrt3)/(3) sqrt3 (2sqrt3)/(3)

Question

For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence).  12,quad8sqrt3,quad16,dots  2sqrt3  (sqrt3)/(3)  sqrt3  (2sqrt3)/(3)

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Check for Difference: To determine whether the sequence is arithmetic or geometric, we need to check if there is a constant difference (arithmetic sequence) or a constant ratio (geometric sequence) between consecutive terms.Subtract First and Second Term: First, let's check for a common difference by subtracting the second term from the first term: 8√3 - 12.Subtract Third and Second Term: Perform the subtraction: 8√3 - 12 = 8(1.732) - 12 ≈ 13.856 - 12 = 1.856. This is not an integer, and we should check the next pair of terms to see if there is a consistent difference.Check for Ratio: Now, subtract the third term from the second term: 16 - 8√3.Divide Second by First Term: Perform the subtraction: 16 - 8√3 = 16 - 8(1.732) ≈ 16 - 13.856 = 2.144. This difference is not the same as the previous one, indicating that the sequence is not arithmetic.Divide Third by Second Term: Since the sequence is not arithmetic, let's check for a common ratio by dividing the second term by the first term: (8√3) / 12.Confirm Geometric Sequence: Perform the division: (8√3) / 12 = (8/12)√3 = (2/3)√3. This simplifies to (2√3) / 3.Now, divide the third term by the second term to check if the ratio is consistent: 16 / (8√3).Perform the division: 16 / (8√3) = (16/8) / √3 = 2 / √3. To rationalize the denominator, multiply the numerator and denominator by √3 to get (2√3) / 3.The ratio between the second and first term is (2√3) / 3, and the ratio between the third and second term is also (2√3) / 3. This indicates that the sequence is geometric with a common ratio of (2√3) / 3.

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