To analyze the sequence [tex]\(-3, 5, -7, 9, -11, \ldots\)[/tex], let's look into its various properties and patterns.
1. Examining if the sequence is arithmetic:
In an arithmetic sequence, the difference between any two consecutive terms is constant. Let's compute the differences between consecutive terms:
- [tex]\(5 - (-3) = 8\)[/tex]
- [tex]\(-7 - 5 = -12\)[/tex]
- [tex]\(9 - (-7) = 16\)[/tex]
- [tex]\(-11 - 9 = -20\)[/tex]
The differences are [tex]\(8, -12, 16, -20\)[/tex]. Since these differences are not constant, the sequence is not arithmetic.
2. Examining if the sequence is geometric:
In a geometric sequence, the ratio between any two consecutive terms is constant. Let's compute the ratios of consecutive terms:
- [tex]\(\frac{5}{-3} \approx -1.67\)[/tex]
- [tex]\(\frac{-7}{5} = -1.4\)[/tex]
- [tex]\(\frac{9}{-7} \approx -1.29\)[/tex]
- [tex]\(\frac{-11}{9} \approx -1.22\)[/tex]
The ratios are approximately [tex]\(-1.67, -1.4, -1.29, -1.22\)[/tex]. Since these ratios are not constant, the sequence is not geometric.
3. Identifying other patterns:
Let's examine other patterns in the sequence. Notably, the sequence alternates in signs:
- The first term is negative [tex]\((-3)\)[/tex].
- The second term is positive [tex]\((5)\)[/tex].
- The third term is negative [tex]\((-7)\)[/tex].
- The fourth term is positive [tex]\((9)\)[/tex].
- The fifth term is negative [tex]\((-11)\)[/tex].
We can observe a pattern where the signs of the terms alternate between positive and negative consistently. Thus, the sequence has alternating signs.
Based on this analysis, the sequence [tex]\(-3, 5, -7, 9, -11, \ldots\)[/tex]:
- is not arithmetic,
- is not geometric, and
- has alternating signs.
So, the statements that correctly describe the sequence are:
- The sequence is not arithmetic.
- The sequence is not geometric.
- The sequence has alternating signs.
