For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence). $\newline$ $\sqrt{7}, \quad 7, \quad 7 \sqrt{7}, \ldots$ $\newline$ $\frac{\sqrt{7}}{7}$ $\newline$ $\sqrt{7}$ $\newline$ $\frac{2 \sqrt{7}}{7}$ $\newline$ $2 \sqrt{7}$

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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

\sqrt{7}, \quad 7, \quad 7 \sqrt{7}, \ldots

\newline

\frac{\sqrt{7}}{7}

\newline

\sqrt{7}

\newline

\frac{2 \sqrt{7}}{7}

\newline

2 \sqrt{7}

Identify type of sequence: Identify the type of sequence by examining the relationship between consecutive terms.
Check arithmetic sequence: The given sequence is: $\sqrt{7}$ , $7$ , $7\sqrt{7}$ , ... To determine if it's an arithmetic sequence, we need to check if the difference between consecutive terms is constant.
Calculate difference: Calculate the difference between the second and the first term: $7 - \sqrt{7}$ .
Compare differences: Calculate the difference between the third and the second term: $7\sqrt{7} - 7$ .
Check geometric sequence: Compare the differences. If they are equal, it's an arithmetic sequence; if not, it's not an arithmetic sequence.
Calculate ratio: The difference between the second and the first term is $7 - \sqrt{7}$ , which is not a simple rational number. $\newline$ The difference between the third and the second term is $7\sqrt{7} - 7$ , which simplifies to $6\sqrt{7}$ . $\newline$ Since $7 - \sqrt{7}$ is not equal to $6\sqrt{7}$ , the sequence is not arithmetic.
Simplify ratio: Now, let's check if it's a geometric sequence by finding the ratio between consecutive terms.
Calculate ratio: Calculate the ratio between the second and the first term: $\frac{7}{\sqrt{7}}$ .
Simplify ratio: Simplify the ratio: $\frac{7}{\sqrt{7}} = \sqrt{7}.$
Compare ratios: Calculate the ratio between the third and the second term: $\frac{7\sqrt{7}}{7}.$
Compare ratios: Calculate the ratio between the third and the second term: $\frac{7\sqrt{7}}{7}$ . Simplify the ratio: $\frac{7\sqrt{7}}{7} = \sqrt{7}$ .
Compare ratios: Calculate the ratio between the third and the second term: $\frac{7\sqrt{7}}{7}$ . Simplify the ratio: $\frac{7\sqrt{7}}{7} = \sqrt{7}$ . Compare the ratios. If they are equal, it's a geometric sequence; if not, it's not a geometric sequence.
Compare ratios: Calculate the ratio between the third and the second term: $\frac{7\sqrt{7}}{7}$ . Simplify the ratio: $\frac{7\sqrt{7}}{7} = \sqrt{7}$ . Compare the ratios. If they are equal, it's a geometric sequence; if not, it's not a geometric sequence. Since the ratio between the second and the first term is $\sqrt{7}$ and the ratio between the third and the second term is also $\sqrt{7}$ , the sequence is geometric with a common ratio of $\sqrt{7}$ .