For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). $\newline$ $10 \sqrt{2}, \quad 10, \quad 5 \sqrt{2}, \quad \ldots$ $\newline$ $\frac{3 \sqrt{2}}{2}$ $\newline$ $3 \sqrt{2}$ $\newline$ $\frac{\sqrt{2}}{2}$ $\newline$ $\sqrt{2}$

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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

10 \sqrt{2}, \quad 10, \quad 5 \sqrt{2}, \quad \ldots

\newline

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

\newline

3 \sqrt{2}

\newline

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

\newline

\sqrt{2}

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: $\newline$ Common difference = $10 - 10\sqrt{2}$ $\newline$ Since $10$ and $10\sqrt{2}$ 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: $\newline$ Common ratio = $\frac{10}{10\sqrt{2}}$ $\newline$ To simplify, we can multiply the numerator and denominator by $\sqrt{2}$ to rationalize the denominator: $\newline$ Common ratio = $\frac{(10 \times \sqrt{2})}{(10\sqrt{2} \times \sqrt{2})} = \frac{\sqrt{2}}{2}$
Conclusion: Now, let's verify this common ratio by dividing the third term by the second term: $\newline$ Common ratio = $\frac{5\sqrt{2}}{10}$ $\newline$ To simplify, we divide both the numerator and denominator by $5$ : $\newline$ Common ratio = $\left(\frac{\sqrt{2}}{2}\right)$
Conclusion: Now, let's verify this common ratio by dividing the third term by the second term: $\newline$ Common ratio = $\frac{5\sqrt{2}}{10}$ $\newline$ To simplify, we divide both the numerator and denominator by $5$ : $\newline$ Common ratio = $\left(\frac{\sqrt{2}}{2}\right)$ 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 $\frac{\sqrt{2}}{2}$ .