For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). 1,quadsqrt3,quad3,quad dots (sqrt3)/(3) (2sqrt3)/(3) sqrt3 2sqrt3

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).  1,quadsqrt3,quad3,quad dots  (sqrt3)/(3)  (2sqrt3)/(3)  sqrt3  2sqrt3

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Sequence Type Determination: To determine if the sequence is arithmetic or geometric, we need to examine the pattern of the terms. An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms.Arithmetic Sequence Check: Let's check for an arithmetic sequence by subtracting the second term from the first term and the third term from the second term: Second term - First term = √3 - 1 Third term - Second term = 3 - √3Arithmetic Sequence Calculation: Now, let's calculate the differences: √3 - 1 ≈ 0.732 3 - √3 ≈ 1.268 These differences are not equal, so the sequence is not arithmetic.Geometric Sequence Check: Next, let's check for a geometric sequence by dividing the second term by the first term and the third term by the second term: Second term / First term = (√3) / 1 = √3 Third term / Second term = 3 / √3Geometric Sequence Calculation: Now, let's calculate the ratios: √3 ≈ 1.732 3 / √3 = 3 / (√3) * (√3/√3) = (3√3) / 3 = √3 The ratios are equal, so the sequence is geometric.Common Ratio Determination: Since the sequence is geometric, the common ratio is the value we found by dividing any term by the previous term, which is √3.

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