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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).\newline7,7,77,… \sqrt{7}, \quad 7, \quad 7 \sqrt{7}, \quad \ldots \newline77 \frac{\sqrt{7}}{7} \newline7 \sqrt{7} \newline277 \frac{2 \sqrt{7}}{7} \newline27 2 \sqrt{7}

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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).\newline7,7,77,… \sqrt{7}, \quad 7, \quad 7 \sqrt{7}, \quad \ldots \newline77 \frac{\sqrt{7}}{7} \newline7 \sqrt{7} \newline277 \frac{2 \sqrt{7}}{7} \newline27 2 \sqrt{7}
  1. 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.\newlineCalculation: 7−77 - \sqrt{7}
  2. Check Second Difference: The difference between the first two terms is 7−77 - \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.\newlineCalculation: 77−77\sqrt{7} - 7
  3. Check Common Ratio: The difference between the second and third terms is 77−77\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.\newlineCalculation: 77\frac{7}{\sqrt{7}}
  4. Check Second Ratio: The ratio between the second and first terms is 77\frac{7}{\sqrt{7}}, which simplifies to 7\sqrt{7}. Let's check the ratio between the third and second terms to see if it is the same.\newlineCalculation: 777\frac{7\sqrt{7}}{7}
  5. Sequence is Geometric: The ratio between the third and second terms is (777)(\frac{7\sqrt{7}}{7}), which simplifies to 7\sqrt{7}. Since the ratio between consecutive terms is the same, the sequence is geometric with a common ratio of 7\sqrt{7}.

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