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What kind of sequence is this?\newline156,144,132,120,…156, 144, 132, 120, \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?\newline156,144,132,120,…156, 144, 132, 120, \dots\newlineChoices:\newline(A) arithmetic\newline(B) geometric\newline(C) both\newline(D) neither
  1. Verify Consecutive Differences: Let's verify if the differences between consecutive terms are uniform. Given sequence: 156,144,132,120,…156, 144, 132, 120, \ldots\newlineAre the consecutive differences in the sequence equal? \newline144−156=−12144 - 156 = -12, 132−144=−12132 - 144 = -12, 120−132=−12120 - 132 = -12.\newlineThe consecutive differences in the sequence are equal and the common difference is −12-12.
  2. Identify Arithmetic Sequence: Since the sequence has a common difference, it is an arithmetic sequence. \newlineNow, let's check if it could also be a geometric sequence by finding the ratios between consecutive terms.\newline144156=0.923076923…\frac{144}{156} = 0.923076923\ldots, 132144=0.916666666…\frac{132}{144} = 0.916666666\ldots, 120132=0.909090909…\frac{120}{132} = 0.909090909\ldots\newlineThe ratios between consecutive terms are not equal.
  3. Check Geometric Sequence: An arithmetic sequence has a common difference between consecutive terms, while a geometric sequence has a common ratio between consecutive terms.\newlineSince the given sequence has a common difference but not a common ratio, it is an arithmetic sequence and not a geometric sequence.

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