To classify the series \( 12 + (-8) + \frac{16}{3} + \left(-\frac{32}{9}\right) + \frac{64}{27} + \ldots \), we need to identify a pattern in the terms to determine if it's arithmetic or geometric.
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Check for Arithmetic Series: In an arithmetic series, the difference between consecutive terms is constant. Let's calculate the differences between the first few terms:
- The difference between the first term \( 12 \) and the second term \( -8 \): \[ -8 - 12 = -20 \]
- The difference between the second term \( -8 \) and the third term \( \frac{16}{3} \): \[ \frac{16}{3} - (-8) = \frac{16}{3} + \frac{24}{3} = \frac{40}{3} \approx 13.33 \]
- The difference keeps changing, so it is not arithmetic.
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Check for Geometric Series: In a geometric series, the ratio between consecutive terms must be constant. Let's find the ratios:
- The ratio of the second term \( -8 \) to the first term \( 12 \): \[ \frac{-8}{12} = -\frac{2}{3} \]
- The ratio of the third term \( \frac{16}{3} \) to the second term \( -8 \): \[ \frac{\frac{16}{3}}{-8} = \frac{16}{3} \cdot \frac{-1}{8} = -\frac{2}{3} \]
- The ratio of the fourth term \( -\frac{32}{9} \) to the third term \( \frac{16}{3} \): \[ \frac{-\frac{32}{9}}{\frac{16}{3}} = -\frac{32}{9} \cdot \frac{3}{16} = -\frac{2}{3} \]
- The ratio of the fifth term \( \frac{64}{27} \) to the fourth term \( -\frac{32}{9} \): \[ \frac{\frac{64}{27}}{-\frac{32}{9}} = \frac{64}{27} \cdot \frac{-9}{32} = -\frac{2}{3} \]
Since the ratio between consecutive terms is consistently \(-\frac{2}{3}\), we can confirm that the series is geometric.
- Convergence/Divergence of the Geometric Series: A geometric series converges if the absolute value of the common ratio \( r \) is less than 1. In this case, \( r = -\frac{2}{3} \), and its absolute value is: \[ |r| = \frac{2}{3} < 1 \] Thus, the series converges.
In conclusion, the series is classified as geometric and is convergent.
The correct response is: geometric, convergent