To determine whether the given sequence is geometric and to find the common ratio if it is, follow these steps:
1. Identify the given sequence: 2, -12, 72, -432, ...
2. Calculate the common ratio for the first few terms:
- First term ([tex]\(a_1\)[/tex]): 2
- Second term ([tex]\(a_2\)[/tex]): -12
- Third term ([tex]\(a_3\)[/tex]): 72
- Fourth term ([tex]\(a_4\)[/tex]): -432
3. Compute the ratio between each consecutive pair of terms:
- [tex]\(\frac{a_2}{a_1} = \frac{-12}{2} = -6\)[/tex]
- [tex]\(\frac{a_3}{a_2} = \frac{72}{-12} = -6\)[/tex]
- [tex]\(\frac{a_4}{a_3} = \frac{-432}{72} = -6\)[/tex]
4. Verify if the ratios are consistent: The common ratio we calculated between each pair of consecutive terms ([tex]\(\frac{a_2}{a_1}\)[/tex], [tex]\(\frac{a_3}{a_2}\)[/tex], and [tex]\(\frac{a_4}{a_3}\)[/tex]) is the same, which is -6.
5. Conclusion:
Since the common ratio is the same for each pair of consecutive terms, the sequence is indeed geometric.
6. Identify the common ratio:
The common ratio ([tex]\(r\)[/tex]) is -6.
Final answer:
OA. The given sequence is geometric. The common ratio is [tex]\(r = -6\)[/tex].
