For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). $\newline$ $2 \sqrt{2}, \quad 6, \quad 9 \sqrt{2}, \quad \ldots$ $\newline$ $\frac{\sqrt{2}}{2}$ $\newline$ $3 \sqrt{2}$ $\newline$ $\sqrt{2}$ $\newline$ $\frac{3 \sqrt{2}}{2}$

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Precalculus

Identify arithmetic and geometric series

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Q. For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence).

\newline

2 \sqrt{2}, \quad 6, \quad 9 \sqrt{2}, \quad \ldots

\newline

\frac{\sqrt{2}}{2}

\newline

3 \sqrt{2}

\newline

\sqrt{2}

\newline

\frac{3 \sqrt{2}}{2}

Identify Sequence Type: First, let's identify the type of sequence by examining the relationship between consecutive terms. $\newline$ We have the sequence: $2\sqrt{2}, 6, 9\sqrt{2}, \ldots$ $\newline$ To determine if it's an arithmetic sequence, we look for a common difference by subtracting each term from the next one. $\newline$ Let's subtract the first term from the second term: $6 - 2\sqrt{2}$ .
Check for Arithmetic Sequence: Now, we calculate the difference: $6 - 2\sqrt{2} = 6 - 2 \times 1.414... \approx 6 - 2.828... \approx 3.172...$ $\newline$ This does not result in a rational number, and it seems unlikely that the sequence is arithmetic because the terms involve $\sqrt{2}$ , 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: $\frac{6}{2\sqrt{2}}$ .
Check for Geometric Sequence: We calculate the ratio: $\frac{6}{2\sqrt{2}} = \frac{3}{\sqrt{2}}$ . To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{2}$ : $\frac{3\sqrt{2}}{2}$ .
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: $(9\sqrt{2}) / 6$ .
Confirm Geometric Sequence: We calculate the ratio: $(9\sqrt{2}) / 6 = (3\sqrt{2}) / 2$ , which is the same as the ratio we found between the first and second terms. $\newline$ This confirms that the sequence is geometric with a common ratio of $(3\sqrt{2}) / 2$ .