Arithmetic And Geometric Sequences Coloring Activity - Free Printable

Let’s solve each part step by step.

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## Part 1: Classify each sequence as arithmetic, geometric, or neither

Definitions:

- Arithmetic sequence: Each term is obtained by adding a constant difference to the previous term.
→ Check if `a₂ - a₁ = a₃ - a₂ = ...`

- Geometric sequence: Each term is obtained by multiplying the previous term by a constant ratio.
→ Check if `a₂/a₁ = a₃/a₂ = ...` (and terms ≠ 0)

- Neither: If neither condition holds.

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1) –3, –18, –108, –648, –3888, ...

Check ratios:

- –18 / –3 = 6
- –108 / –18 = 6
- –648 / –108 = 6
- –3888 / –648 = 6

✔ Constant ratio = 6 → Geometric

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2) 2, 4, 12, 48, 240, ...

Check ratios:

- 4/2 = 2
- 12/4 = 3
- 48/12 = 4
- 240/48 = 5

Ratios: 2, 3, 4, 5 → increasing → Not constant

Check differences:

- 4–2=2, 12–4=8, 48–12=36, 240–48=192 → not constant

→ Neither

*(Note: This is actually factorial-like: 2×1=2, 2×2=4, 4×3=12, 12×4=48, 48×5=240 → next would be ×6, etc. But it's not arithmetic or geometric.)*

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3) –35, 165, 365, 565, 765, ...

Check differences:

- 165 – (–35) = 200
- 365 – 165 = 200
- 565 – 365 = 200
- 765 – 565 = 200

✔ Constant difference = 200 → Arithmetic

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4) –2, 6, –18, 54, –162, ...

Check ratios:

- 6 / –2 = –3
- –18 / 6 = –3
- 54 / –18 = –3
- –162 / 54 = –3

✔ Constant ratio = –3 → Geometric

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5) –7, 93, 193, 293, 393, ...

Check differences:

- 93 – (–7) = 100
- 193 – 93 = 100
- 293 – 193 = 100
- 393 – 293 = 100

✔ Constant difference = 100 → Arithmetic

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6) 8, 14, 20, 26, 32, ...

Differences:

- 14–8=6, 20–14=6, 26–20=6, 32–26=6

✔ Constant difference = 6 → Arithmetic

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7) –1, –2, –6, –24, –120, ...

Check ratios:

- –2 / –1 = 2
- –6 / –2 = 3
- –24 / –6 = 4
- –120 / –24 = 5

Ratios: 2, 3, 4, 5 → increasing → Not constant

Differences: –1, –4, –18, –96 → not constant

→ Neither

*(This is factorial: –1×2=–2, –2×3=–6, –6×4=–24, –24×5=–120 → so it's recursive but not arithmetic/geometric.)*

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8) –18, –12, –6, 0, 6, ...

Differences:

- –12 – (–18) = 6
- –6 – (–12) = 6
- 0 – (–6) = 6
- 6 – 0 = 6

✔ Constant difference = 6 → Arithmetic

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9) –2.5, –5, –10, –20, –40, ...

Ratios:

- –5 / –2.5 = 2
- –10 / –5 = 2
- –20 / –10 = 2
- –40 / –20 = 2

✔ Constant ratio = 2 → Geometric

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10) 1, 2, 6, 24, 120, ...

Ratios:

- 2/1 = 2
- 6/2 = 3
- 24/6 = 4
- 120/24 = 5

→ Ratios: 2,3,4,5 → not constant

Differences: 1,4,18,96 → not constant

→ Neither

*(Again, this is factorial: 1×2=2, 2×3=6, 6×4=24, 24×5=120 → recursive, not arithmetic/geometric.)*

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## ✔ Part 1 Summary:

1) Geometric
2) Neither
3) Arithmetic
4) Geometric
5) Arithmetic
6) Arithmetic
7) Neither
8) Arithmetic
9) Geometric
10) Neither

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## Part 2: Find the next three terms in each sequence

We’ll analyze patterns.

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11) 2, 4, 12, 48, 240, ...

Pattern: Multiply by increasing integers.

- 2 × 2 = 4
- 4 × 3 = 12
- 12 × 4 = 48
- 48 × 5 = 240
→ Next: 240 × 6 = 1440, ×7 = 10080, ×8 = 80640

✔ Next three: 1440, 10080, 80640

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12) 2, 5, 10, 17, 26, ...

Differences:

- 5–2=3
- 10–5=5
- 17–10=7
- 26–17=9

→ Differences are odd numbers: +3, +5, +7, +9 → next: +11, +13, +15

So:

- 26 + 11 = 37
- 37 + 13 = 50
- 50 + 15 = 65

✔ Next three: 37, 50, 65

*(Also: these are n² + 1: 1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17, 5²+1=26 → 6²+1=37, etc.)*

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13) 1, 9, 25, 49, 81, ...

These are squares of odd numbers:

- 1² = 1
- 3² = 9
- 5² = 25
- 7² = 49
- 9² = 81

Next: 11² = 121, 13² = 169, 15² = 225

✔ Next three: 121, 169, 225

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14) 4, 16, 36, 64, 100, ...

Squares of even numbers:

- 2² = 4
- 4² = 16
- 6² = 36
- 8² = 64
- 10² = 100

Next: 12² = 144, 14² = 196, 16² = 256

✔ Next three: 144, 196, 256

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15) –6, –2, 0, 1, 3/2, ...

Let’s look at differences:

- –2 – (–6) = 4
- 0 – (–2) = 2
- 1 – 0 = 1
- 3/2 – 1 = 0.5

Differences: 4, 2, 1, 0.5 → halving each time!

Next differences: 0.25, 0.125, 0.0625

So:

- 3/2 + 0.25 = 1.5 + 0.25 = 1.75 = 7/4
- 7/4 + 0.125 = 1.75 + 0.125 = 1.875 = 15/8
- 15/8 + 0.0625 = 1.875 + 0.0625 = 1.9375 = 31/16

✔ Next three: 7/4, 15/8, 31/16

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16) –9, –7, –4, 0, 5, ...

Differences:

- –7 – (–9) = 2
- –4 – (–7) = 3
- 0 – (–4) = 4
- 5 – 0 = 5

→ Differences: +2, +3, +4, +5 → next: +6, +7, +8

So:

- 5 + 6 = 11
- 11 + 7 = 18
- 18 + 8 = 26

✔ Next three: 11, 18, 26

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17) –2, –4, –12, –48, –240, ...

Pattern: Multiply by increasing integers (starting from 2):

- –2 × 2 = –4
- –4 × 3 = –12
- –12 × 4 = –48
- –48 × 5 = –240

Next: ×6, ×7, ×8

- –240 × 6 = –1440
- –1440 × 7 = –10080
- –10080 × 8 = –80640

✔ Next three: –1440, –10080, –80640

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18) 6, 8, 11, 15, 20, ...

Differences:

- 8–6=2
- 11–8=3
- 15–11=4
- 20–15=5

→ Differences: +2, +3, +4, +5 → next: +6, +7, +8

So:

- 20 + 6 = 26
- 26 + 7 = 33
- 33 + 8 = 41

✔ Next three: 26, 33, 41

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19) –30, –14, –6, –2, 0, ...

Differences:

- –14 – (–30) = 16
- –6 – (–14) = 8
- –2 – (–6) = 4
- 0 – (–2) = 2

→ Differences: 16, 8, 4, 2 → halving each time!

Next differences: 1, 0.5, 0.25

So:

- 0 + 1 = 1
- 1 + 0.5 = 1.5
- 1.5 + 0.25 = 1.75

✔ Next three: 1, 1.5, 1.75 (or 1, 3/2, 7/4)

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20) –2, 1, 6, 13, 22, ...

Differences:

- 1 – (–2) = 3
- 6 – 1 = 5
- 13 – 6 = 7
- 22 – 13 = 9

→ Differences: +3, +5, +7, +9 → odd numbers increasing → next: +11, +13, +15

So:

- 22 + 11 = 33
- 33 + 13 = 46
- 46 + 15 = 61

✔ Next three: 33, 46, 61

*(Also: n² – 3: 1²–3=–2, 2²–3=1, 3²–3=6, 4²–3=13, 5²–3=22 → 6²–3=33, etc.)*

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## ✔ Final Answers:

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Part 1: Classification

1) Geometric
2) Neither
3) Arithmetic
4) Geometric
5) Arithmetic
6) Arithmetic
7) Neither
8) Arithmetic
9) Geometric
10) Neither

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Part 2: Next Three Terms

11) 1440, 10080, 80640
12) 37, 50, 65
13) 121, 169, 225
14) 144, 196, 256
15) 7/4, 15/8, 31/16
16) 11, 18, 26
17) –1440, –10080, –80640
18) 26, 33, 41
19) 1, 1.5, 1.75 (or 1, 3/2, 7/4)
20) 33, 46, 61

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Let me know if you’d like explanations for any specific pattern!