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

sqrt7,quad7,quad7sqrt7,quad dots

(sqrt7)/(7)

sqrt7

(2sqrt7)/(7)

2sqrt7

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

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Function transformation rules

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

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

\newline

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

\newline

\sqrt{7}

\newline

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

\newline

2 \sqrt{7}

Check Common Difference: To determine whether the sequence is arithmetic or geometric, we need to examine the relationship between consecutive terms. Let's first check if there is a common difference by subtracting the first term from the second term. $\newline$ Calculation: $7 - \sqrt{7}$
Check Second Difference: The difference between the first two terms is $7 - \sqrt{7}$ , which is not a simple rational number. Let's check the difference between the second and third terms to see if it is the same. $\newline$ Calculation: $7\sqrt{7} - 7$
Check Common Ratio: The difference between the second and third terms is $7\sqrt{7} - 7$ , which is also not a simple rational number and is different from the difference between the first two terms. This suggests that the sequence is not arithmetic. Now, let's check if there is a common ratio by dividing the second term by the first term. $\newline$ Calculation: $\frac{7}{\sqrt{7}}$
Check Second Ratio: The ratio between the second and first terms is $\frac{7}{\sqrt{7}}$ , which simplifies to $\sqrt{7}$ . Let's check the ratio between the third and second terms to see if it is the same. $\newline$ Calculation: $\frac{7\sqrt{7}}{7}$
Sequence is Geometric: The ratio between the third and second terms is $(\frac{7\sqrt{7}}{7})$ , which simplifies to $\sqrt{7}$ . Since the ratio between consecutive terms is the same, the sequence is geometric with a common ratio of $\sqrt{7}$ .

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