" Classify the series. sum_(n = 0)^12 (n + 2)^3 Choices: [[arithmetic][geometric][both][neither]] "

Calculate first term: Calculate the first term by substituting n = 0 into (n + 2)^3. (0 + 2)^3 = 2^3 = 8.Calculate second term: Calculate the second term by substituting n = 1 into (n + 2)^3. (1 + 2)^3 = 3^3 = 27.Calculate third term: Calculate the third term by substituting n = 2 into (n + 2)^3. (2 + 2)^3 = 4^3 = 64.Check for arithmetic progression: Check for arithmetic progression by finding the difference between consecutive terms. Second term - First term = 27 - 8 = 19. Third term - Second term = 64 - 27 = 37. The differences are not the same; hence, not arithmetic.Check for geometric progression: Check for geometric progression by finding the ratio between consecutive terms. Second term / First term = 27 / 8. Third term / Second term = 64 / 27. The ratios are not the same; hence, not geometric.Classification as neither: Since the series is neither arithmetic nor geometric, the classification is ＂neither＂.

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

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Math Problems

Algebra 2

Identify arithmetic and geometric series

Full solution

Q. Classify the series.

\sum_{n = 0}^{12} (n + 2)^3

\newline

Choices:

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Calculate first term: Calculate the first term by substituting $n = 0$ into $(n + 2)^3$ . $\$0 + 2$ ^ $3$ = $2$ ^ $3$ = $8$ \).
Calculate second term: Calculate the second term by substituting $n = 1$ into $(n + 2)^3$ . $\$1 + 2$ ^ $3$ = $3$ ^ $3$ = $27$ \).
Calculate third term: Calculate the third term by substituting $n = 2$ into $(n + 2)^3$ . $\$2 + 2$ ^ $3$ = $4$ ^ $3$ = $64$ \).
Check for arithmetic progression: Check for arithmetic progression by finding the difference between consecutive terms. $\newline$ Second term - First term = $27 - 8 = 19$ . $\newline$ Third term - Second term = $64 - 27 = 37$ . $\newline$ The differences are not the same; hence, not arithmetic.
Check for geometric progression: Check for geometric progression by finding the ratio between consecutive terms. $\newline$ Second term / First term = $\frac{27}{8}$ . $\newline$ Third term / Second term = $\frac{64}{27}$ . $\newline$ The ratios are not the same; hence, not geometric.
Classification as neither: Since the series is neither arithmetic nor geometric, the classification is ＂neither＂.