To find the first five terms of the sequence given by \( a_n = (-2)^n \):
- Calculate each term:
- 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:
\[ -2, 4, -8, 16, -32 \]
-
Determine if the sequence is arithmetic or geometric:
- A sequence is arithmetic if the difference between consecutive terms is constant.
- \( 4 - (-2) = 4 + 2 = 6 \)
- \( -8 - 4 = -12 \)
- \( 16 - (-8) = 16 + 8 = 24 \)
- \( -32 - 16 = -48 \)
The differences are not constant, so the sequence is not arithmetic.
- A sequence is geometric if the ratio of consecutive terms is constant.
- \( \frac{4}{-2} = -2 \)
- \( \frac{-8}{4} = -2 \)
- \( \frac{16}{-8} = -2 \)
- \( \frac{-32}{16} = -2 \)
The ratios are constant, so the sequence is geometric with a common ratio of \( -2 \).
- A sequence is arithmetic if the difference between consecutive terms is constant.
-
Conclusion: The correct option is: Option #3: The sequence is geometric, and the common ratio is -2.