Math worksheet designed to help students practice identifying arithmetic sequences and calculating the next terms in a series.

Let’s go through each sequence one by one. We’ll check if it’s arithmetic — that means the difference between consecutive terms is always the same. If it is, we’ll find the next three terms by adding that common difference again and again.

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1. 17, 12, 7, 2, ...

Check differences:
- 12 - 17 = -5
- 7 - 12 = -5
- 2 - 7 = -5

✔ Arithmetic! Common difference = -5

Next three terms:
- 2 + (-5) = -3
- -3 + (-5) = -8
- -8 + (-5) = -13

→ Next: -3, -8, -13

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2. -129, -98, -67, ...

Differences:
- -98 - (-129) = -98 + 129 = 31
- -67 - (-98) = -67 + 98 = 31

✔ Arithmetic! Common difference = 31

Next three:
- -67 + 31 = -36
- -36 + 31 = -5
- -5 + 31 = 26

→ Next: -36, -5, 26

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3. 1/4, 2/6, 3/8, 4/10, ...

First, simplify fractions to see pattern:
- 1/4 = 0.25
- 2/6 = 1/3 ≈ 0.333...
- 3/8 = 0.375
- 4/10 = 2/5 = 0.4

Now check differences:
- 1/3 - 1/4 = (4 - 3)/12 = 1/12
- 3/8 - 1/3 = (9 - 8)/24 = 1/24
- 2/5 - 3/8 = (16 - 15)/40 = 1/40

Not the same → ✘ Not arithmetic.

Wait — maybe look at numerators and denominators separately?

Numerators: 1, 2, 3, 4 → increasing by 1
Denominators: 4, 6, 8, 10 → increasing by 2

So general term: n / (2n + 2)? Let’s test:

For n=1: 1/(2+2)=1/4 ✔
n=2: 2/(4+2)=2/6 ✔
n=3: 3/(6+2)=3/8 ✔
n=4: 4/(8+2)=4/10 ✔

But this is not an *arithmetic* sequence — because the actual values don’t increase by a constant amount. The differences are changing (as we saw: 1/12, 1/24, 1/40).

So even though there’s a pattern, it’s not arithmetic.

→ ✘ Not arithmetic.

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4. 12, 24, 36, 48, ...

Differences:
- 24 - 12 = 12
- 36 - 24 = 12
- 48 - 36 = 12

✔ Arithmetic! Common difference = 12

Next three:
- 48 + 12 = 60
- 60 + 12 = 72
- 72 + 12 = 84

→ Next: 60, 72, 84

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5. 36k, 49k, 64k, 81k, ...

These look like squares times k:
- 36k = 6²k
- 49k = 7²k
- 64k = 8²k
- 81k = 9²k

So next would be 10²k = 100k, etc.

But is it arithmetic? Check differences:

- 49k - 36k = 13k
- 64k - 49k = 15k
- 81k - 64k = 17k

Differences: 13k, 15k, 17k → increasing by 2k each time → NOT constant.

✘ Not arithmetic.

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6. 3, -6, 12, -24, ...

Check ratios first (since signs alternate and numbers double):

- -6 / 3 = -2
- 12 / -6 = -2
- -24 / 12 = -2

This is geometric (multiplying by -2), not arithmetic.

Check differences:
- -6 - 3 = -9
- 12 - (-6) = 18
- -24 - 12 = -36

Not constant → ✘ Not arithmetic.

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7. -29, -2, 25, ...

Differences:
- -2 - (-29) = -2 + 29 = 27
- 25 - (-2) = 25 + 2 = 27

✔ Arithmetic! Common difference = 27

Next three:
- 25 + 27 = 52
- 52 + 27 = 79
- 79 + 27 = 106

→ Next: 52, 79, 106

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8. -4.5, -9.5, -14.5, ...

Differences:
- -9.5 - (-4.5) = -9.5 + 4.5 = -5
- -14.5 - (-9.5) = -14.5 + 9.5 = -5

✔ Arithmetic! Common difference = -5

Next three:
- -14.5 + (-5) = -19.5
- -19.5 + (-5) = -24.5
- -24.5 + (-5) = -29.5

→ Next: -19.5, -24.5, -29.5

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9. 7, -3.5, -14, ...

Check differences:
- -3.5 - 7 = -10.5
- -14 - (-3.5) = -14 + 3.5 = -10.5

✔ Arithmetic! Common difference = -10.5

Next three:
- -14 + (-10.5) = -24.5
- -24.5 + (-10.5) = -35
- -35 + (-10.5) = -45.5

→ Next: -24.5, -35, -45.5

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10. 8, -2, 1/2, ...

Check differences:
- -2 - 8 = -10
- 0.5 - (-2) = 0.5 + 2 = 2.5

Not the same → ✘ Not arithmetic.

Check ratios:
- -2 / 8 = -0.25
- 0.5 / -2 = -0.25

Geometric with ratio -1/4 → not arithmetic.

→ ✘ Not arithmetic.

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Final Answer:

1. Yes; next three: -3, -8, -13
2. Yes; next three: -36, -5, 26
3. No
4. Yes; next three: 60, 72, 84
5. No
6. No
7. Yes; next three: 52, 79, 106
8. Yes; next three: -19.5, -24.5, -29.5
9. Yes; next three: -24.5, -35, -45.5
10. No