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What kind of sequence is this?\newline7,21,84,420,…7, 21, 84, 420, \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?\newline7,21,84,420,…7, 21, 84, 420, \dots\newlineChoices:\newline(A) arithmetic\newline(B) geometric\newline(C) both\newline(D) neither
  1. Check Arithmetic Differences: Let's first check if the sequence is arithmetic by finding the differences between consecutive terms. Given sequence: 7,21,84,420,…7, 21, 84, 420, \ldots\newlineCalculate the differences: 21−7=1421 - 7 = 14, 84−21=6384 - 21 = 63, 420−84=336420 - 84 = 336.
  2. Verify Equality of Differences: Now, let's see if these differences are equal. Are the consecutive differences in the sequence equal? 1414, 6363, and 336336 are not equal.\newlineThis means the sequence is not arithmetic.
  3. Check Geometric Ratios: Next, let's check if the sequence is geometric by finding the ratios between consecutive terms. Given sequence: 7,21,84,420,…7, 21, 84, 420, \ldots\newlineCalculate the ratios: 217=3\frac{21}{7} = 3, 8421=4\frac{84}{21} = 4, 42084=5\frac{420}{84} = 5.
  4. Verify Equality of Ratios: Now, let's see if these ratios are equal. Are the ratios between consecutive terms in the sequence equal? 33, 44, and 55 are not equal.\newlineThis means the sequence is not geometric.
  5. Identify Sequence Type: Since the sequence is neither arithmetic (no common difference) nor geometric (no common ratio), the correct choice is (D)(D) neither.

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