Arithmetic Sequences Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each problem one by one, step by step.

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Problem 1: Is the sequence arithmetic or not? {8, 16, 32, 64...}

An arithmetic sequence has a constant difference between terms. Let’s check:

- 16 - 8 = 8
- 32 - 16 = 16 → Not the same as 8
- 64 - 32 = 32 → Also different

Since the differences are not the same, this is not an arithmetic sequence.

✔ Final Answer for #1: Not arithmetic

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Problem 2: Find the next 3 terms in the sequence. {-5, -1, 3, 7, 11...}

First, find the common difference:

- (-1) - (-5) = 4
- 3 - (-1) = 4
- 7 - 3 = 4
- 11 - 7 = 4 → Common difference is +4

So add 4 to the last term (11):

- 11 + 4 = 15
- 15 + 4 = 19
- 19 + 4 = 23

✔ Final Answer for #2: 15, 19, 23

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Problem 3: Is the sequence finite or infinite? {14, 9, 4, -1, -6...}

The “...” at the end means it continues forever — so it’s infinite.

Also, even though we can see a pattern (subtracting 5), there’s no stopping point shown.

✔ Final Answer for #3: Infinite

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Problem 4: What is the common difference in this sequence? {8.2, 1.8, -4.6, -11}

Find the difference between consecutive terms:

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

All differences are -6.4 → That’s the common difference.

✔ Final Answer for #4: -6.4

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Problem 5: Find the next 5 terms in the sequence. {16, 25, 34...}

Find the common difference:

- 25 - 16 = 9
- 34 - 25 = 9 → So add 9 each time

Next 5 terms:

- 34 + 9 = 43
- 43 + 9 = 52
- 52 + 9 = 61
- 61 + 9 = 70
- 70 + 9 = 79

✔ Final Answer for #5: 43, 52, 61, 70, 79

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Problem 6: The first term is 15. The common difference is -4. Write the first 5 terms.

Start with 15, then subtract 4 each time:

- Term 1: 15
- Term 2: 15 - 4 = 11
- Term 3: 11 - 4 = 7
- Term 4: 7 - 4 = 3
- Term 5: 3 - 4 = -1

✔ Final Answer for #6: 15, 11, 7, 3, -1

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

Recursive rule means: tell how to get the next term from the previous one.

Common difference: 14 - 7 = 7 → so each term is previous term + 7

Also need to state the first term.

Recursive rule:
- a₁ = 7
- aₙ = aₙ₋₁ + 7

✔ Final Answer for #7:
a₁ = 7; aₙ = aₙ₋₁ + 7

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

Explicit rule: formula using n directly.

First, find common difference:

- 16 - 3 = 13
- 29 - 16 = 13 → d = 13

First term a₁ = 3

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

Simplify:
aₙ = 3 + 13n - 13 = 13n - 10

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

✔ Final Answer for #8:
Explicit rule: aₙ = 13n - 10; a₂₄ = 302

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

Find common difference:

- 67 - 94 = -27
- 40 - 67 = -27 → d = -27

First term a₁ = 94

Explicit formula:
aₙ = a₁ + (n - 1)d
→ aₙ = 94 + (n - 1)(-27)

Simplify:
aₙ = 94 - 27(n - 1)
= 94 - 27n + 27
= 121 - 27n

Check with n=1: 121 - 27(1) = 94 ✔
n=2: 121 - 54 = 67 ✔
n=3: 121 - 81 = 40 ✔

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

✔ Final Answer for #9:
Explicit rule: aₙ = 121 - 27n; a₇₅ = -1904

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

Wait — let’s check if this is arithmetic.

Differences:

- 16 - (-4) = 20
- 36 - 16 = 20 → Yes! Common difference is 20

So recursive rule:

- a₁ = -4
- aₙ = aₙ₋₁ + 20

Now find a₆:

We can build up:

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

Or use explicit formula to verify:
aₙ = -4 + (n-1)(20) = 20n - 24
a₆ = 20(6) - 24 = 120 - 24 = 96 ✔

✔ Final Answer for #10:
Recursive rule: a₁ = -4; aₙ = aₙ₋₁ + 20; a₆ = 96

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Final Answers Summary:

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