Examples of linear and non-linear sequences in mathematics.

Problem Analysis

The task is to classify sequences into linear sequences and non-linear sequences. A linear sequence is one where the difference between consecutive terms is constant. In contrast, a non-linear sequence does not have a constant difference between consecutive terms.

Let's analyze each sequence step by step.

---

Linear Sequences

#### Sequence 1: \(8, 16, 24, 32, 40, \ldots\)
- Difference between consecutive terms:
\[
16 - 8 = 8, \quad 24 - 16 = 8, \quad 32 - 24 = 8, \quad 40 - 32 = 8
\]
- The difference is constant (\(8\)).
- Conclusion: This is a linear sequence.

#### Sequence 2: \(7, 15, 23, 31, 39, \ldots\)
- Difference between consecutive terms:
\[
15 - 7 = 8, \quad 23 - 15 = 8, \quad 31 - 23 = 8, \quad 39 - 31 = 8
\]
- The difference is constant (\(8\)).
- Conclusion: This is a linear sequence.

#### Sequence 3: \(-3, -10, -17, -24, \ldots\)
- Difference between consecutive terms:
\[
-10 - (-3) = -7, \quad -17 - (-10) = -7, \quad -24 - (-17) = -7
\]
- The difference is constant (\(-7\)).
- Conclusion: This is a linear sequence.

#### Sequence 4: \(31, 32, 33, 34, \ldots\)
- Difference between consecutive terms:
\[
32 - 31 = 1, \quad 33 - 32 = 1, \quad 34 - 33 = 1
\]
- The difference is constant (\(1\)).
- Conclusion: This is a linear sequence.

#### Sequence 5: \(7, 2.5, -2, -6.5, \ldots\)
- Difference between consecutive terms:
\[
2.5 - 7 = -4.5, \quad -2 - 2.5 = -4.5, \quad -6.5 - (-2) = -4.5
\]
- The difference is constant (\(-4.5\)).
- Conclusion: This is a linear sequence.

#### Sequence 6: \(a, a + 3, a + 6, \ldots\)
- Difference between consecutive terms:
\[
(a + 3) - a = 3, \quad (a + 6) - (a + 3) = 3
\]
- The difference is constant (\(3\)).
- Conclusion: This is a linear sequence.

---

Non-Linear Sequences

#### Sequence 1: \(2, 5, 9, 14, 20, \ldots\)
- Difference between consecutive terms:
\[
5 - 2 = 3, \quad 9 - 5 = 4, \quad 14 - 9 = 5, \quad 20 - 14 = 6
\]
- The differences are not constant (\(3, 4, 5, 6\)).
- Conclusion: This is a non-linear sequence.

#### Sequence 2: \(7, -7, 7, -7, 7, \ldots\)
- Difference between consecutive terms:
\[
-7 - 7 = -14, \quad 7 - (-7) = 14, \quad -7 - 7 = -14, \quad 7 - (-7) = 14
\]
- The differences alternate between \(-14\) and \(14\), which is not constant.
- Conclusion: This is a non-linear sequence.

#### Sequence 3: \(2, 4, 6, 8, 11, \ldots\)
- Difference between consecutive terms:
\[
4 - 2 = 2, \quad 6 - 4 = 2, \quad 8 - 6 = 2, \quad 11 - 8 = 3
\]
- The differences are not constant (\(2, 2, 2, 3\)).
- Conclusion: This is a non-linear sequence.

#### Sequence 4: \(100, 80, 70, 65, \ldots\)
- Difference between consecutive terms:
\[
80 - 100 = -20, \quad 70 - 80 = -10, \quad 65 - 70 = -5
\]
- The differences are not constant (\(-20, -10, -5\)).
- Conclusion: This is a non-linear sequence.

#### Sequence 5: \(1, 4, 9, 16, \ldots\)
- Difference between consecutive terms:
\[
4 - 1 = 3, \quad 9 - 4 = 5, \quad 16 - 9 = 7
\]
- The differences are not constant (\(3, 5, 7\)).
- Conclusion: This is a non-linear sequence.

---

Final Classification

- Linear Sequences:
\[
8, 16, 24, 32, 40, \ldots
\]
\[
7, 15, 23, 31, 39, \ldots
\]
\[
-3, -10, -17, -24, \ldots
\]
\[
31, 32, 33, 34, \ldots
\]
\[
7, 2.5, -2, -6.5, \ldots
\]
\[
a, a + 3, a + 6, \ldots
\]

- Non-Linear Sequences:
\[
2, 5, 9, 14, 20, \ldots
\]
\[
7, -7, 7, -7, 7, \ldots
\]
\[
2, 4, 6, 8, 11, \ldots
\]
\[
100, 80, 70, 65, \ldots
\]
\[
1, 4, 9, 16, \ldots
\]

---

Final Answer

\[
\boxed{
\text{Linear Sequences: } 8, 16, 24, 32, 40, \ldots; \, 7, 15, 23, 31, 39, \ldots; \, -3, -10, -17, -24, \ldots; \, 31, 32, 33, 34, \ldots; \, 7, 2.5, -2, -6.5, \ldots; \, a, a + 3, a + 6, \ldots
}
\]
\[
\boxed{
\text{Non-Linear Sequences: } 2, 5, 9, 14, 20, \ldots; \, 7, -7, 7, -7, 7, \ldots; \, 2, 4, 6, 8, 11, \ldots; \, 100, 80, 70, 65, \ldots; \, 1, 4, 9, 16, \ldots
}
\]