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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,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}, \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}, \ldots \newline77 \frac{\sqrt{7}}{7} \newline7 \sqrt{7} \newline277 \frac{2 \sqrt{7}}{7} \newline27 2 \sqrt{7}
  1. Identify type of sequence: Identify the type of sequence by examining the relationship between consecutive terms.
  2. Check arithmetic sequence: The given sequence is: 7\sqrt{7}, 77, 777\sqrt{7}, ... To determine if it's an arithmetic sequence, we need to check if the difference between consecutive terms is constant.
  3. Calculate difference: Calculate the difference between the second and the first term: 7−77 - \sqrt{7}.
  4. Compare differences: Calculate the difference between the third and the second term: 77−77\sqrt{7} - 7.
  5. Check geometric sequence: Compare the differences. If they are equal, it's an arithmetic sequence; if not, it's not an arithmetic sequence.
  6. Calculate ratio: The difference between the second and the first term is 7−77 - \sqrt{7}, which is not a simple rational number.\newlineThe difference between the third and the second term is 77−77\sqrt{7} - 7, which simplifies to 676\sqrt{7}.\newlineSince 7−77 - \sqrt{7} is not equal to 676\sqrt{7}, the sequence is not arithmetic.
  7. Simplify ratio: Now, let's check if it's a geometric sequence by finding the ratio between consecutive terms.
  8. Calculate ratio: Calculate the ratio between the second and the first term: 77\frac{7}{\sqrt{7}}.
  9. Simplify ratio: Simplify the ratio: 77=7.\frac{7}{\sqrt{7}} = \sqrt{7}.
  10. Compare ratios: Calculate the ratio between the third and the second term: 777.\frac{7\sqrt{7}}{7}.
  11. Compare ratios: Calculate the ratio between the third and the second term: 777\frac{7\sqrt{7}}{7}. Simplify the ratio: 777=7\frac{7\sqrt{7}}{7} = \sqrt{7}.
  12. Compare ratios: Calculate the ratio between the third and the second term: 777\frac{7\sqrt{7}}{7}. Simplify the ratio: 777=7\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.
  13. Compare ratios: Calculate the ratio between the third and the second term: 777\frac{7\sqrt{7}}{7}. Simplify the ratio: 777=7\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 7\sqrt{7} and the ratio between the third and the second term is also 7\sqrt{7}, the sequence is geometric with a common ratio of 7\sqrt{7}.

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