Let's first calculate the first five terms of the sequence defined by \( a_n = (-2)^n \):
- For \( n = 1 \): \[ a_1 = (-2)^1 = -2 \]
- For \( n = 2 \): \[ a_2 = (-2)^2 = 4 \]
- For \( n = 3 \): \[ a_3 = (-2)^3 = -8 \]
- For \( n = 4 \): \[ a_4 = (-2)^4 = 16 \]
- For \( n = 5 \): \[ a_5 = (-2)^5 = -32 \]
So, the first five terms of the sequence are:
- \( a_1 = -2 \)
- \( a_2 = 4 \)
- \( a_3 = -8 \)
- \( a_4 = 16 \)
- \( a_5 = -32 \)
Now, let's determine whether the sequence is arithmetic or geometric.
A sequence is arithmetic if the difference between consecutive terms is constant. Calculating the differences:
- \( 4 - (-2) = 6 \)
- \( -8 - 4 = -12 \)
- \( 16 - (-8) = 24 \)
- \( -32 - 16 = -48 \)
Since the differences are not constant, the sequence is not arithmetic.
A sequence is geometric if the ratio between consecutive terms is constant. Calculating the ratios:
- \( \frac{4}{-2} = -2 \)
- \( \frac{-8}{4} = -2 \)
- \( \frac{16}{-8} = -2 \)
- \( \frac{-32}{16} = -2 \)
Since the ratios are constant and equal to \(-2\), the sequence is geometric.
Therefore, the correct option is:
Option #3: The sequence is geometric, and the common ratio is \(-2\).