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Classify the series. ∑n=012(n+2)3 \sum_{n = 0}^{12} (n + 2)^3 \newlineChoices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}

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Q. Classify the series. ∑n=012(n+2)3 \sum_{n = 0}^{12} (n + 2)^3 \newlineChoices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
  1. Calculate first term: Calculate the first term by substituting n=0n = 0 into (n+2)3(n + 2)^3.$0+2\$0 + 2^33 = 22^33 = 88\).
  2. Calculate second term: Calculate the second term by substituting n=1n = 1 into (n+2)3(n + 2)^3.$1+2\$1 + 2^33 = 33^33 = 2727\).
  3. Calculate third term: Calculate the third term by substituting n=2n = 2 into (n+2)3(n + 2)^3.$2+2\$2 + 2^33 = 44^33 = 6464\).
  4. Check for arithmetic progression: Check for arithmetic progression by finding the difference between consecutive terms.\newlineSecond term - First term = 27−8=1927 - 8 = 19.\newlineThird term - Second term = 64−27=3764 - 27 = 37.\newlineThe differences are not the same; hence, not arithmetic.
  5. Check for geometric progression: Check for geometric progression by finding the ratio between consecutive terms.\newlineSecond term / First term = 278\frac{27}{8}.\newlineThird term / Second term = 6427\frac{64}{27}.\newlineThe ratios are not the same; hence, not geometric.
  6. Classification as neither: Since the series is neither arithmetic nor geometric, the classification is ＂neither＂.

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