Comprehensive worksheet designed to help algebra students master the basics of arithmetic sequences through varied practice problems.

Let's solve each problem step by step and explain the reasoning.

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1. Is the sequence arithmetic or not?

Sequence: {8, 16, 32, 64...}

- Check the difference between consecutive terms:
- 16 - 8 = 8
- 32 - 16 = 16
- 64 - 32 = 32

The differences are not constant (8, 16, 32), so this is not an arithmetic sequence.

✔ Answer: No, it is not arithmetic.

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2. Find the next 3 terms in the sequence.

Sequence: {-5, -1, 3, 7, 11...}

- Find the common difference:
- -1 - (-5) = 4
- 3 - (-1) = 4
- 7 - 3 = 4
- 11 - 7 = 4

Common difference = +4

Add 4 to get next terms:
- 11 + 4 = 15
- 15 + 4 = 19
- 19 + 4 = 23

✔ Answer: 15, 19, 23

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3. Is the sequence finite or infinite?

Sequence: {14, 9, 4, -1, -6...}

- The sequence has "..." at the end, which means it continues indefinitely.
- Also, there's a pattern: subtracting 5 each time.

✔ Answer: Infinite

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4. What is the common difference in this sequence?

Sequence: {8.2, 1.8, -4.6, -11}

- 1.8 - 8.2 = -6.4
- -4.6 - 1.8 = -6.4
- -11 - (-4.6) = -6.4

✔ Answer: Common difference = -6.4

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5. Find the next 5 terms in the sequence.

Sequence: {16, 25, 34...}

- Difference: 25 - 16 = 9
- 34 - 25 = 9 → Common difference = +9

Next terms:
- 34 + 9 = 43
- 43 + 9 = 52
- 52 + 9 = 61
- 61 + 9 = 70
- 70 + 9 = 79

✔ Answer: 43, 52, 61, 70, 79

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6. The first term is 15. Common difference is -4. Write the first 5 terms.

Use:
a₁ = 15
a₂ = 15 + (-4) = 11
a₃ = 11 + (-4) = 7
a₄ = 7 + (-4) = 3
a₅ = 3 + (-4) = -1

✔ Answer: 15, 11, 7, 3, -1

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7. Write a recursive rule for the nth term of the sequence: {7, 14, 21, 28...}

- First term: a₁ = 7
- Common difference: 14 - 7 = 7 → d = 7

Recursive rule:
- a₁ = 7
- aₙ = aₙ₋₁ + 7 for n ≥ 2

✔ Answer:
a₁ = 7
aₙ = aₙ₋₁ + 7

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8. Write an explicit rule for the nth term of the sequence: {3, 16, 29...}. Then find a₂₄.

- a₁ = 3
- d = 16 - 3 = 13

Explicit formula for arithmetic sequence:
aₙ = a₁ + (n - 1)d
aₙ = 3 + (n - 1)(13)
aₙ = 3 + 13n - 13
aₙ = 13n - 10

Now find a₂₄:
a₂₄ = 13(24) - 10 = 312 - 10 = 302

✔ Answer:
Explicit rule: aₙ = 13n - 10
a₂₄ = 302

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9. Write an explicit rule for the nth term of the sequence: {94, 67, 40...}. Then find a₇₅.

- a₁ = 94
- d = 67 - 94 = -27

Explicit rule:
aₙ = a₁ + (n - 1)d
aₙ = 94 + (n - 1)(-27)
aₙ = 94 - 27(n - 1)
aₙ = 94 - 27n + 27
aₙ = 121 - 27n

Now find a₇₅:
a₇₅ = 121 - 27(75) = 121 - 2025 = -1904

✔ Answer:
Explicit rule: aₙ = 121 - 27n
a₇₅ = -1904

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10. Write a recursive rule for the nth term of the sequence: {-4, 16, 36...}. Then find a₆

First, check if it's arithmetic:
- 16 - (-4) = 20
- 36 - 16 = 20 → Yes, common difference = +20

So it’s arithmetic.

Recursive rule:
- a₁ = -4
- aₙ = aₙ₋₁ + 20 for n ≥ 2

Now find a₆:

- a₁ = -4
- a₂ = -4 + 20 = 16
- a₃ = 16 + 20 = 36
- a₄ = 36 + 20 = 56
- a₅ = 56 + 20 = 76
- a₆ = 76 + 20 = 96

✔ Answer:
Recursive rule:
a₁ = -4
aₙ = aₙ₋₁ + 20
a₆ = 96

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | Not arithmetic |
| 2 | 15, 19, 23 |
| 3 | Infinite |
| 4 | -6.4 |
| 5 | 43, 52, 61, 70, 79 |
| 6 | 15, 11, 7, 3, -1 |
| 7 | a₁ = 7; aₙ = aₙ₋₁ + 7 |
| 8 | aₙ = 13n - 10; a₂₄ = 302 |
| 9 | aₙ = 121 - 27n; a₇₅ = -1904 |
| 10 | a₁ = -4; aₙ = aₙ₋₁ + 20; a₆ = 96 |

Let me know if you'd like these written neatly on paper format!