For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). 2sqrt2,quad6,quad9sqrt2,quad dots (sqrt2)/(2) 3sqrt2 sqrt2 (3sqrt2)/(2)

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).  2sqrt2,quad6,quad9sqrt2,quad dots  (sqrt2)/(2)  3sqrt2  sqrt2  (3sqrt2)/(2)

Bytelearn · Accepted Answer

Identify Sequence Type: First, let's identify the type of sequence by examining the relationship between consecutive terms. We have the sequence: 2√2, 6, 9√2, ... To determine if it's an arithmetic sequence, we look for a common difference by subtracting each term from the next one. Let's subtract the first term from the second term: 6 - 2√2.Check for Arithmetic Sequence: Now, we calculate the difference: 6 - 2√2 = 6 - 2 * 1.414... ≈ 6 - 2.828... ≈ 3.172... This does not result in a rational number, and it seems unlikely that the sequence is arithmetic because the terms involve √2, which suggests a possible geometric pattern.Calculate Difference: Next, we check if it's a geometric sequence by finding a common ratio. We divide the second term by the first term: 6 / (2√2).Check for Geometric Sequence: We calculate the ratio: 6 / (2√2) = 3 / √2. To rationalize the denominator, we multiply the numerator and denominator by √2: (3√2) / 2.Calculate Common Ratio: Now, let's check if this ratio holds for the next pair of terms. We divide the third term by the second term: (9√2) / 6.Confirm Geometric Sequence: We calculate the ratio: (9√2) / 6 = (3√2) / 2, which is the same as the ratio we found between the first and second terms. This confirms that the sequence is geometric with a common ratio of (3√2) / 2.

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