For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). $\newline$ $1, \quad \sqrt{3}, \quad 3, \quad \ldots$ $\newline$ $\frac{\sqrt{3}}{3}$ $\newline$ $\frac{2 \sqrt{3}}{3}$ $\newline$ $\sqrt{3}$ $\newline$ $2 \sqrt{3}$

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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

1, \quad \sqrt{3}, \quad 3, \quad \ldots

\newline

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

\newline

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

\newline

\sqrt{3}

\newline

2 \sqrt{3}

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: $\newline$ Second term - First term = $\sqrt{3} - 1$ $\newline$ Third term - Second term = $3 - \sqrt{3}$
Arithmetic Sequence Calculation: Now, let's calculate the differences: $\newline$ $\sqrt{3} - 1 \approx 0.732$ $\newline$ $3 - \sqrt{3} \approx 1.268$ $\newline$ 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: $\newline$ Second term / First term = $\frac{\sqrt{3}}{1}$ = $\sqrt{3}$ $\newline$ Third term / Second term = $\frac{3}{\sqrt{3}}$
Geometric Sequence Calculation: Now, let's calculate the ratios: $\newline$ $\sqrt{3} \approx 1.732$ $\newline$ $\frac{3}{\sqrt{3}} = \frac{3}{(\sqrt{3})} * (\frac{\sqrt{3}}{\sqrt{3}}) = \frac{(3\sqrt{3})}{3} = \sqrt{3}$ $\newline$ 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 $\sqrt{3}$ .