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What kind of sequence is this?\newline69,58,48,39,…69, 58, 48, 39, \dots\newlineChoices:\newline(A) arithmetic\newline(B) geometric\newline(C) both\newline(D) neither

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Q. What kind of sequence is this?\newline69,58,48,39,…69, 58, 48, 39, \dots\newlineChoices:\newline(A) arithmetic\newline(B) geometric\newline(C) both\newline(D) neither
  1. Check for Arithmetic Sequence: Let's first check if the sequence is arithmetic by finding the differences between consecutive terms. Given sequence: 69,58,48,39,…69, 58, 48, 39, \ldots\newlineCalculate the differences: 58−69=−1158 - 69 = -11, 48−58=−1048 - 58 = -10, 39−48=−939 - 48 = -9.
  2. Analyze Differences: Now, let's analyze the differences. Are the differences between consecutive terms equal? −11-11, −10-10, −9-9. The differences are not equal; they are decreasing by 11 each time.
  3. Check for Geometric Sequence: Since the differences are not constant, the sequence is not arithmetic. Now, let's check if it's geometric by finding the ratios between consecutive terms.\newlineCalculate the ratios: 5869\frac{58}{69}, 4858\frac{48}{58}, 3948\frac{39}{48}. We can see that these ratios are not the same just by looking at the numbers, but let's calculate them for confirmation.
  4. Calculate Ratios: Calculate the exact ratios: 5869≈0.84\frac{58}{69} \approx 0.84, 4858≈0.83\frac{48}{58} \approx 0.83, 3948≈0.81\frac{39}{48} \approx 0.81. These ratios are not equal.
  5. Sequence Classification: Since the sequence has neither a common difference nor a common ratio, it is neither arithmetic nor geometric.

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