Arithmetic And Geometric Sequences Coloring Activity - Free Printable

Let's solve each part of the problem step by step.

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Part 1: Determine if each sequence is arithmetic, geometric, or neither

A sequence is:
- Arithmetic if the common difference (difference between consecutive terms) is constant.
- Geometric if the common ratio (ratio of consecutive terms) is constant.
- Neither if neither condition holds.

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#### 1) –3, –18, –108, –648, –3888, ...

Check ratios:
- $ \frac{-18}{-3} = 6 $
- $ \frac{-108}{-18} = 6 $
- $ \frac{-648}{-108} = 6 $
- $ \frac{-3888}{-648} = 6 $

✔ Common ratio = 6 → Geometric

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

Check ratios:
- $ \frac{4}{2} = 2 $
- $ \frac{12}{4} = 3 $
- $ \frac{48}{12} = 4 $
- $ \frac{240}{48} = 5 $

Ratio changes → Not geometric

Check differences:
- 4 - 2 = 2
- 12 - 4 = 8
- 48 - 12 = 36
- 240 - 48 = 192

No common difference → Neither

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#### 3) –35, 165, 365, 565, 765, ...

Check differences:
- 165 - (-35) = 200
- 365 - 165 = 200
- 565 - 365 = 200
- 765 - 565 = 200

✔ Common difference = 200 → Arithmetic

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#### 4) –2, 6, –18, 54, –162, ...

Check ratios:
- $ \frac{6}{-2} = -3 $
- $ \frac{-18}{6} = -3 $
- $ \frac{54}{-18} = -3 $
- $ \frac{-162}{54} = -3 $

✔ Common ratio = –3 → Geometric

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#### 5) –7, 93, 193, 293, 393, ...

Differences:
- 93 - (-7) = 100
- 193 - 93 = 100
- 293 - 193 = 100
- 393 - 293 = 100

✔ Common 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

✔ Common difference = 6 → Arithmetic

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#### 7) –1, –2, –6, –24, –120, ...

Ratios:
- $ \frac{-2}{-1} = 2 $
- $ \frac{-6}{-2} = 3 $
- $ \frac{-24}{-6} = 4 $
- $ \frac{-120}{-24} = 5 $

Ratio increases → Not geometric

Differences:
- –2 - (–1) = –1
- –6 - (–2) = –4
- –24 - (–6) = –18
- –120 - (–24) = –96

No pattern → Neither

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#### 8) –18, –12, –6, 0, 6, ...

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

✔ Common difference = 6 → Arithmetic

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#### 9) –2.5, –5, –10, –20, –40, ...

Ratios:
- $ \frac{-5}{-2.5} = 2 $
- $ \frac{-10}{-5} = 2 $
- $ \frac{-20}{-10} = 2 $
- $ \frac{-40}{-20} = 2 $

✔ Common ratio = 2 → Geometric

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

Ratios:
- $ \frac{2}{1} = 2 $
- $ \frac{6}{2} = 3 $
- $ \frac{24}{6} = 4 $
- $ \frac{120}{24} = 5 $

Ratio increases → Not geometric

Differences:
- 2 - 1 = 1
- 6 - 2 = 4
- 24 - 6 = 18
- 120 - 24 = 96

Not constant → Neither

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✔ Summary for Part 1:

| # | Sequence | Type |
|---|--------|------|
| 1 | –3, –18, –108, ... | Geometric |
| 2 | 2, 4, 12, 48, ... | Neither |
| 3 | –35, 165, 365, ... | Arithmetic |
| 4 | –2, 6, –18, 54, ... | Geometric |
| 5 | –7, 93, 193, ... | Arithmetic |
| 6 | 8, 14, 20, 26, ... | Arithmetic |
| 7 | –1, –2, –6, –24, ... | Neither |
| 8 | –18, –12, –6, 0, ... | Arithmetic |
| 9 | –2.5, –5, –10, ... | Geometric |
|10 | 1, 2, 6, 24, 120, ... | Neither |

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Part 2: Find the next three terms in each sequence

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

Look at ratios:
- $ \frac{4}{2} = 2 $
- $ \frac{12}{4} = 3 $
- $ \frac{48}{12} = 4 $
- $ \frac{240}{48} = 5 $

So the multiplier increases by 1 each time: ×2, ×3, ×4, ×5,...

Next multipliers: ×6, ×7, ×8

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

✔ Next three terms: 1440, 10080, 80640

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

Look at differences:
- 5 - 2 = 3
- 10 - 5 = 5
- 17 - 10 = 7
- 26 - 17 = 9

Differences: 3, 5, 7, 9 → increasing by 2 → next: 11, 13, 15

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

✔ Next three terms: 37, 50, 65

Note: This sequence is $ n^2 + 1 $:
n=1: 1+1=2, n=2: 4+1=5, n=3: 9+1=10, etc.

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

These are squares of odd numbers:
- $1^2 = 1$
- $3^2 = 9$
- $5^2 = 25$
- $7^2 = 49$
- $9^2 = 81$

Next: $11^2 = 121$, $13^2 = 169$, $15^2 = 225$

✔ Next three: 121, 169, 225

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

These are squares of even numbers:
- $2^2 = 4$
- $4^2 = 16$
- $6^2 = 36$
- $8^2 = 64$
- $10^2 = 100$

Next: $12^2 = 144$, $14^2 = 196$, $16^2 = 256$

✔ Next three: 144, 196, 256

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#### 15) –6, –2, 0, 1, 3/2, ...

Convert to decimals: –6, –2, 0, 1, 1.5

Differences:
- –2 - (–6) = 4
- 0 - (–2) = 2
- 1 - 0 = 1
- 1.5 - 1 = 0.5

Pattern: 4, 2, 1, 0.5 → halving each time

So next differences: 0.25, 0.125, 0.0625

Add to last term:
- 1.5 + 0.25 = 1.75 = 7/4
- 7/4 + 1/8 = 14/8 + 1/8 = 15/8
- 15/8 + 1/16 = 30/16 + 1/16 = 31/16

✔ Next three: 7/4, 15/8, 31/16

Alternatively, this is a recursive pattern where each term increases by half the previous increment.

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#### 16) –9, –7, –4, 0, 5, ...

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

So differences: 2, 3, 4, 5 → increasing by 1

Next: +6, +7, +8

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

✔ Next three: 11, 18, 26

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#### 17) –2, –4, –12, –48, –240, ...

Ratios:
- $ \frac{-4}{-2} = 2 $
- $ \frac{-12}{-4} = 3 $
- $ \frac{-48}{-12} = 4 $
- $ \frac{-240}{-48} = 5 $

So multiplying by 2, 3, 4, 5 → 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

So differences: 2, 3, 4, 5 → next: 6, 7, 8

- 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

So differences: 16, 8, 4, 2 → halving each time

Next: 1, 0.5, 0.25

- 0 + 1 = 1
- 1 + 0.5 = 1.5 = 3/2
- 1.5 + 0.25 = 1.75 = 7/4

✔ Next three: 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 by 2

Next: 11, 13, 15

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

✔ Next three: 33, 46, 61

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✔ Final Answers:

#### Part 1: Type of Sequence
1. Geometric
2. Neither
3. Arithmetic
4. Geometric
5. Arithmetic
6. Arithmetic
7. Neither
8. Arithmetic
9. Geometric
10. Neither

#### Part 2: Next Three Terms
11. 1440, 10080, 80640
12. 37, 50, 65
13. 121, 169, 225
14. 144, 196, 256
15. $ \frac{7}{4}, \frac{15}{8}, \frac{31}{16} $
16. 11, 18, 26
17. –1440, –10080, –80640
18. 26, 33, 41
19. 1, $ \frac{3}{2} $, $ \frac{7}{4} $
20. 33, 46, 61

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