To determine which sequences are arithmetic, let's first recall what characterizes an arithmetic sequence. A sequence is arithmetic if the difference between consecutive terms is constant throughout the sequence. This difference is called the common difference.
Let’s analyze each given sequence step by step:
1. Sequence 1: [tex]\(-5, 5, -5, 5, -5, \ldots\)[/tex]
- Examine the differences between consecutive terms:
- [tex]\(5 - (-5) = 10\)[/tex]
- [tex]\(-5 - 5 = -10\)[/tex]
- [tex]\(5 - (-5) = 10\)[/tex]
- [tex]\(-5 - 5 = -10\)[/tex]
- The differences are not consistent (alternating between 10 and -10), so this sequence is not arithmetic.
2. Sequence 2: [tex]\(96, 48, 24, 12, 6\)[/tex]
- Examine the differences between consecutive terms:
- [tex]\(48 - 96 = -48\)[/tex]
- [tex]\(24 - 48 = -24\)[/tex]
- [tex]\(12 - 24 = -12\)[/tex]
- [tex]\(6 - 12 = -6\)[/tex]
- The differences are not consistent, so this sequence is not arithmetic.
3. Sequence 3: [tex]\(18, 5.5, -7, -19.5, -32, \ldots\)[/tex]
- Examine the differences between consecutive terms:
- [tex]\(5.5 - 18 = -12.5\)[/tex]
- [tex]\(-7 - 5.5 = -12.5\)[/tex]
- [tex]\(-19.5 - (-7) = -12.5\)[/tex]
- [tex]\(-32 - (-19.5) = -12.5\)[/tex]
- The differences are consistent ([tex]\(-12.5\)[/tex]), so this sequence is arithmetic.
4. Sequence 4: [tex]\(-1, -3, -9, -27, -81, \ldots\)[/tex]
- Examine the differences between consecutive terms:
- [tex]\(-3 - (-1) = -2\)[/tex]
- [tex]\(-9 - (-3) = -6\)[/tex]
- [tex]\(-27 - (-9) = -18\)[/tex]
- [tex]\(-81 - (-27) = -54\)[/tex]
- The differences are not consistent, so this sequence is not arithmetic.
5. Sequence 5: [tex]\(16, 32, 48, 64, 80\)[/tex]
- Examine the differences between consecutive terms:
- [tex]\(32 - 16 = 16\)[/tex]
- [tex]\(48 - 32 = 16\)[/tex]
- [tex]\(64 - 48 = 16\)[/tex]
- [tex]\(80 - 64 = 16\)[/tex]
- The differences are consistent ([tex]\(16\)[/tex]), so this sequence is arithmetic.
Therefore, the sequences that are arithmetic are:
[tex]\[ 18, 5.5, -7, -19.5, -32, \ldots \][/tex]
and
[tex]\[ 16, 32, 48, 64, 80. \][/tex]
