Arithmetic sequences worksheet with 12 problems for students to solve.

Problem Analysis:

The task involves working with arithmetic sequences. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference.

#### Key Concepts:
1. Arithmetic Sequence: A sequence where each term differs from the previous term by a constant value (common difference).
2. General Term Formula: For an arithmetic sequence, the \( n \)-th term is given by:
\[
a_n = a_1 + (n-1)d
\]
where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

---

Solution to Each Part:

#### Part 1: Determine whether each sequence is arithmetic. If so, find the common difference.

1. Sequence 1: \( 15.2, 8.5, 1.8, -4.9, \ldots \)
- Calculate the differences between consecutive terms:
\[
8.5 - 15.2 = -6.7, \quad 1.8 - 8.5 = -6.7, \quad -4.9 - 1.8 = -6.7
\]
- The differences are constant (\( -6.7 \)).
- Conclusion: This is an arithmetic sequence with a common difference of \( d = -6.7 \).

2. Sequence 2: \( 8.4, 16.1, 23.8, 31.5, \ldots \)
- Calculate the differences between consecutive terms:
\[
16.1 - 8.4 = 7.7, \quad 23.8 - 16.1 = 7.7, \quad 31.5 - 23.8 = 7.7
\]
- The differences are constant (\( 7.7 \)).
- Conclusion: This is an arithmetic sequence with a common difference of \( d = 7.7 \).

3. Sequence 3: \( 5.1, -0.8, -6.7, -12.6, \ldots \)
- Calculate the differences between consecutive terms:
\[
-0.8 - 5.1 = -5.9, \quad -6.7 - (-0.8) = -5.9, \quad -12.6 - (-6.7) = -5.9
\]
- The differences are constant (\( -5.9 \)).
- Conclusion: This is an arithmetic sequence with a common difference of \( d = -5.9 \).

4. Sequence 4: \( 14.3, 20.3, 26.3, 32.3, \ldots \)
- Calculate the differences between consecutive terms:
\[
20.3 - 14.3 = 6, \quad 26.3 - 20.3 = 6, \quad 32.3 - 26.3 = 6
\]
- The differences are constant (\( 6 \)).
- Conclusion: This is an arithmetic sequence with a common difference of \( d = 6 \).

---

#### Part 2: Find the first four terms and stated term given the arithmetic sequence, with \( a_n \) as the \( n \)-th term.

5. Sequence 5: \( a_n = 25.5 - 6.8n \), find \( a_{10} \)
- General formula: \( a_n = 25.5 - 6.8n \)
- First four terms:
\[
a_1 = 25.5 - 6.8(1) = 25.5 - 6.8 = 18.7
\]
\[
a_2 = 25.5 - 6.8(2) = 25.5 - 13.6 = 11.9
\]
\[
a_3 = 25.5 - 6.8(3) = 25.5 - 20.4 = 5.1
\]
\[
a_4 = 25.5 - 6.8(4) = 25.5 - 27.2 = -1.7
\]
- \( a_{10} \):
\[
a_{10} = 25.5 - 6.8(10) = 25.5 - 68 = -42.5
\]
- First four terms: \( 18.7, 11.9, 5.1, -1.7 \)
- \( a_{10} \): \( -42.5 \)

6. Sequence 6: \( a_n = 25 - 10n \), find \( a_6 \)
- General formula: \( a_n = 25 - 10n \)
- First four terms:
\[
a_1 = 25 - 10(1) = 25 - 10 = 15
\]
\[
a_2 = 25 - 10(2) = 25 - 20 = 5
\]
\[
a_3 = 25 - 10(3) = 25 - 30 = -5
\]
\[
a_4 = 25 - 10(4) = 25 - 40 = -15
\]
- \( a_6 \):
\[
a_6 = 25 - 10(6) = 25 - 60 = -35
\]
- First four terms: \( 15, 5, -5, -15 \)
- \( a_6 \): \( -35 \)

7. Sequence 7: \( a_n = 11 + 9n \), find \( a_8 \)
- General formula: \( a_n = 11 + 9n \)
- First four terms:
\[
a_1 = 11 + 9(1) = 11 + 9 = 20
\]
\[
a_2 = 11 + 9(2) = 11 + 18 = 29
\]
\[
a_3 = 11 + 9(3) = 11 + 27 = 38
\]
\[
a_4 = 11 + 9(4) = 11 + 36 = 47
\]
- \( a_8 \):
\[
a_8 = 11 + 9(8) = 11 + 72 = 83
\]
- First four terms: \( 20, 29, 38, 47 \)
- \( a_8 \): \( 83 \)

8. Sequence 8: \( a_n = 65 - 35n \), find \( a_7 \)
- General formula: \( a_n = 65 - 35n \)
- First four terms:
\[
a_1 = 65 - 35(1) = 65 - 35 = 30
\]
\[
a_2 = 65 - 35(2) = 65 - 70 = -5
\]
\[
a_3 = 65 - 35(3) = 65 - 105 = -40
\]
\[
a_4 = 65 - 35(4) = 65 - 140 = -75
\]
- \( a_7 \):
\[
a_7 = 65 - 35(7) = 65 - 245 = -180
\]
- First four terms: \( 30, -5, -40, -75 \)
- \( a_7 \): \( -180 \)

---

#### Part 3: Given the first term and common difference, find the first four terms and the formula.

9. Sequence 9: \( a_1 = 25 \), \( d = 100 \)
- General formula: \( a_n = a_1 + (n-1)d \)
\[
a_n = 25 + (n-1)(100)
\]
- First four terms:
\[
a_1 = 25
\]
\[
a_2 = 25 + (2-1)(100) = 25 + 100 = 125
\]
\[
a_3 = 25 + (3-1)(100) = 25 + 200 = 225
\]
\[
a_4 = 25 + (4-1)(100) = 25 + 300 = 325
\]
- First four terms: \( 25, 125, 225, 325 \)
- Formula: \( a_n = 25 + 100(n-1) \)

10. Sequence 10: \( a_1 = 5 \), \( d = 5 \)
- General formula: \( a_n = a_1 + (n-1)d \)
\[
a_n = 5 + (n-1)(5)
\]
- First four terms:
\[
a_1 = 5
\]
\[
a_2 = 5 + (2-1)(5) = 5 + 5 = 10
\]
\[
a_3 = 5 + (3-1)(5) = 5 + 10 = 15
\]
\[
a_4 = 5 + (4-1)(5) = 5 + 15 = 20
\]
- First four terms: \( 5, 10, 15, 20 \)
- Formula: \( a_n = 5 + 5(n-1) \)

11. Sequence 11: \( a_1 = 24 \), \( d = -15 \)
- General formula: \( a_n = a_1 + (n-1)d \)
\[
a_n = 24 + (n-1)(-15)
\]
- First four terms:
\[
a_1 = 24
\]
\[
a_2 = 24 + (2-1)(-15) = 24 - 15 = 9
\]
\[
a_3 = 24 + (3-1)(-15) = 24 - 30 = -6
\]
\[
a_4 = 24 + (4-1)(-15) = 24 - 45 = -21
\]
- First four terms: \( 24, 9, -6, -21 \)
- Formula: \( a_n = 24 - 15(n-1) \)

12. Sequence 12: \( a_1 = 9 \), \( d = -50 \)
- General formula: \( a_n = a_1 + (n-1)d \)
\[
a_n = 9 + (n-1)(-50)
\]
- First four terms:
\[
a_1 = 9
\]
\[
a_2 = 9 + (2-1)(-50) = 9 - 50 = -41
\]
\[
a_3 = 9 + (3-1)(-50) = 9 - 100 = -91
\]
\[
a_4 = 9 + (4-1)(-50) = 9 - 150 = -141
\]
- First four terms: \( 9, -41, -91, -141 \)
- Formula: \( a_n = 9 - 50(n-1) \)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{1. } d = -6.7 & \text{2. } d = 7.7 \\
\text{3. } d = -5.9 & \text{4. } d = 6 \\
\text{5. First four terms: } 18.7, 11.9, 5.1, -1.7, \quad a_{10} = -42.5 & \text{6. First four terms: } 15, 5, -5, -15, \quad a_6 = -35 \\
\text{7. First four terms: } 20, 29, 38, 47, \quad a_8 = 83 & \text{8. First four terms: } 30, -5, -40, -75, \quad a_7 = -180 \\
\text{9. First four terms: } 25, 125, 225, 325, \quad a_n = 25 + 100(n-1) & \text{10. First four terms: } 5, 10, 15, 20, \quad a_n = 5 + 5(n-1) \\
\text{11. First four terms: } 24, 9, -6, -21, \quad a_n = 24 - 15(n-1) & \text{12. First four terms: } 9, -41, -91, -141, \quad a_n = 9 - 50(n-1)
\end{array}
}
\]