For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). $\newline$ $8 \sqrt{3}, \quad 24, \quad 24 \sqrt{3}, \quad \ldots$ $\newline$ $\frac{\sqrt{3}}{3}$ $\newline$ $\sqrt{3}$ $\newline$ $2 \sqrt{3}$ $\newline$ $\frac{2 \sqrt{3}}{3}$

Full solution

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

8 \sqrt{3}, \quad 24, \quad 24 \sqrt{3}, \quad \ldots

\newline

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

\newline

\sqrt{3}

\newline

2 \sqrt{3}

\newline

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

Identify Sequence Type: First, let's identify whether the sequence is arithmetic or geometric. An arithmetic sequence has a common difference between terms, while a geometric sequence has a common ratio. $\newline$ To determine this, we can compare the ratio of consecutive terms. $\newline$ Let's calculate the ratio of the second term to the first term. $\newline$ Ratio = $\frac{24}{8\sqrt{3}}$
Calculate Ratio: Now, let's simplify the ratio. $\newline$ Ratio = $\frac{24}{8\sqrt{3}} = \frac{3}{\sqrt{3}}$
Simplify Ratio: To further simplify, we can rationalize the denominator. $\newline$ Ratio = $(\frac{3}{\sqrt{3}}) \cdot (\frac{\sqrt{3}}{\sqrt{3}}) = \frac{3\sqrt{3}}{3}$
Check Consistency: After simplifying, we get: $\newline$ Ratio = $\frac{3\sqrt{3}}{3} = \sqrt{3}$
Calculate Next Ratio: Now, let's check if this ratio is consistent with the next pair of terms. $\newline$ We calculate the ratio of the third term to the second term. $\newline$ Ratio = $(24\sqrt{3}) / 24$
Simplify Next Ratio: Simplify the ratio. $\newline$ Ratio = $(24\sqrt{3}) / 24 = \sqrt{3}$
Conclude Geometric Sequence: Since the ratio between consecutive terms is consistent and equal to $\sqrt{3}$ , we can conclude that the sequence is geometric with a common ratio of $\sqrt{3}$ .