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What kind of sequence is this?\newline131,118,105,92,…131, 118, 105, 92, \dots\newlineChoices:\newline(A) arithmetic\newline(B) geometric\newline(C) both\newline(D) neither

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Full solution

Q. What kind of sequence is this?\newline131,118,105,92,…131, 118, 105, 92, \dots\newlineChoices:\newline(A) arithmetic\newline(B) geometric\newline(C) both\newline(D) neither
  1. Check Arithmetic Sequence: Let's first check if the sequence is arithmetic by finding the differences between consecutive terms. Given sequence: 131,118,105,92,…131, 118, 105, 92, \ldots\newlineCalculate the differences: 118−131=−13118 - 131 = -13, 105−118=−13105 - 118 = -13, 92−105=−1392 - 105 = -13.
  2. Calculate Common Difference: Since the differences between consecutive terms are equal, we can conclude that the sequence has a common difference of −13-13.
  3. Confirm Arithmetic Sequence: An arithmetic sequence is defined by having a constant difference between its terms. Since we have found a common difference, this sequence is arithmetic.
  4. Check Geometric Sequence: Now, let's check if the sequence could also be geometric by finding the ratios between consecutive terms. Calculate the ratios: 118131\frac{118}{131}, 105118\frac{105}{118}, 92105\frac{92}{105}.
  5. Calculate Ratios: Perform the calculations: 118131≈0.9008\frac{118}{131} \approx 0.9008, 105118≈0.8898\frac{105}{118} \approx 0.8898, 92105≈0.8762\frac{92}{105} \approx 0.8762.
  6. Confirm Not Geometric: Since the ratios between consecutive terms are not equal, the sequence does not have a common ratio and therefore is not a geometric sequence.
  7. Final Conclusion: Based on the calculations, the sequence is arithmetic because it has a common difference, but it is not geometric because it does not have a common ratio.

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