For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). 10sqrt2,quad10,quad5sqrt2,quad dots (3sqrt2)/(2) 3sqrt2 (sqrt2)/(2) sqrt2

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).  10sqrt2,quad10,quad5sqrt2,quad dots  (3sqrt2)/(2)  3sqrt2  (sqrt2)/(2)  sqrt2

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Check for Common Difference: To determine whether the sequence is arithmetic or geometric, we need to check if there is a common difference (which would make it arithmetic) or a common ratio (which would make it geometric). We can do this by comparing consecutive terms.Check for Common Ratio: First, let's check for a common difference by subtracting the second term from the first term: Common difference = 10 - 10√2 Since 10 and 10√2 are not like terms, we cannot simplify this further, indicating that there is no common difference. This suggests that the sequence is not arithmetic.Verify Common Ratio: Next, let's check for a common ratio by dividing the second term by the first term: Common ratio = 10 / 10√2 To simplify, we can multiply the numerator and denominator by √2 to rationalize the denominator: Common ratio = (10 * √2) / (10√2 * √2) = √2 / 2Conclusion: Now, let's verify this common ratio by dividing the third term by the second term: Common ratio = 5√2 / 10 To simplify, we divide both the numerator and denominator by 5: Common ratio = (√2 / 2)Since the common ratio is consistent for the first two pairs of terms, we can conclude that the sequence is geometric with a common ratio of √2 / 2.

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