Patterns Worksheets | Dynamically Created Patterns Worksheets - Free Printable

Let's solve each of these numerical series step by step and explain the pattern.

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1) 4, 8, 10, 20, 22, 44, 46, __, __, __

Step-by-step analysis:
- 4 → 8 (×2)
- 8 → 10 (+2)
- 10 → 20 (×2)
- 20 → 22 (+2)
- 22 → 44 (×2)
- 44 → 46 (+2)

Pattern: Alternating ×2 and +2

So:
- 46 → ? (×2) = 92
- 92 → ? (+2) = 94
- 94 → ? (×2) = 188

✔ Answer: 92, 94, 188

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2) 1, 3, -1, -3, -7, -21, -25, __, __, __

Check differences or patterns:

- 1 → 3 (+2)
- 3 → -1 (-4)
- -1 → -3 (-2)
- -3 → -7 (-4)
- -7 → -21 (-14)
- -21 → -25 (-4)

Wait — maybe two interleaved sequences?

Let’s split into odd and even positions:

Odd positions (1st, 3rd, 5th, 7th):
- 1, -1, -7, -25
- 1 → -1: -2
- -1 → -7: -6
- -7 → -25: -18

Differences: -2, -6, -18 → multiplying by 3 each time?
(-2) × 3 = -6, (-6) × 3 = -18 → next difference: -54
So: -25 + (-54) = -79

Even positions (2nd, 4th, 6th, 8th):
- 3, -3, -21, ?
- 3 → -3: -6
- -3 → -21: -18
- -21 → ? → next should be ×3 in difference: -6, -18, -54 → so -21 -54 = -75

Now let’s list:

Positions:
1. 1
2. 3
3. -1
4. -3
5. -7
6. -21
7. -25
8. ? → even → -75
9. ? → odd → -79
10. ? → even → next even position after -75: difference pattern: -6, -18, -54 → next would be -162 → -75 -162 = -237

But we only need three more terms: positions 8, 9, 10

So:
- 8th: -75
- 9th: -79
- 10th: ?

Wait — but 9th is odd → from earlier: -25 → -79 (difference -54)

Then 10th is even → previous even: -21 → -75 → difference -54 → same as before? But that was -18 to -54 → multiply by 3

From -21 to -75: difference -54
Next even term: -75 - (54×3) = -75 -162 = -237

So sequence continues:
- 8th: -75
- 9th: -79
- 10th: -237

But let’s verify:

Odd terms: 1, -1, -7, -25, -79
Diffs: -2, -6, -18, -54 → ×3 each time ✔

Even terms: 3, -3, -21, -75, -237
Diffs: -6, -18, -54, -162 → ×3 each time ✔

✔ Answer: -75, -79, -237

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3) 0, 1, 1, 2, 3, 5, 8, __, __, __

This is the Fibonacci sequence:
Each term is sum of two preceding ones.

- 0, 1, 1, 2, 3, 5, 8
- Next: 5+8 = 13
- Then: 8+13 = 21
- Then: 13+21 = 34

✔ Answer: 13, 21, 34

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4) 4, 12, 16, 48, 52, 156, 160, __, __, __

Look at pattern:

- 4 → 12 (×3)
- 12 → 16 (+4)
- 16 → 48 (×3)
- 48 → 52 (+4)
- 52 → 156 (×3)
- 156 → 160 (+4)
- 160 → ? (×3) = 480
- 480 → ? (+4) = 484
- 484 → ? (×3) = 1452

Pattern: ×3, +4, ×3, +4, ×3, +4, ...

So alternates between ×3 and +4, starting with ×3.

✔ Answer: 480, 484, 1452

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5) 9, 12, 7, 10, 5, 8, 3, __, __, __

Split into two interleaved sequences:

Odd positions (1st, 3rd, 5th, 7th):
- 9, 7, 5, 3 → decreasing by 2 → next: 1

Even positions (2nd, 4th, 6th, 8th):
- 12, 10, 8, ? → decreasing by 2 → next: 6

Then:
- 9th term: odd → next odd after 3 → 1
- 10th term: even → next even after 8 → 6
- 11th term: odd → after 1 → -1? But wait — let’s see:

Sequence:
1. 9 (odd)
2. 12 (even)
3. 7 (odd)
4. 10 (even)
5. 5 (odd)
6. 8 (even)
7. 3 (odd)
8. ? (even) → 6
9. ? (odd) → 1
10. ? (even) → 4?
Wait — no:

We have:
- Odd: 9, 7, 5, 3 → next: 1
- Even: 12, 10, 8 → next: 6

So:
- 8th: 6
- 9th: 1
- 10th: ? → even → next even after 6 → 4? But pattern: 12→10→8→6 → decreasing by 2 → next: 4

But wait — we need three terms: 8th, 9th, 10th

So:
- 8th: 6
- 9th: 1
- 10th: 4

But 10th is even → yes, 6 → 4

So:
- 8: 6
- 9: 1
- 10: 4

✔ Answer: 6, 1, 4

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6) 16, 22, 19, 25, 22, 28, 25, __, __, __

Look at alternating sequences:

Odd positions (1st, 3rd, 5th, 7th):
- 16, 19, 22, 25 → increasing by 3 → next: 28

Even positions (2nd, 4th, 6th, 8th):
- 22, 25, 28, ? → increasing by 3 → next: 31

So:
- 8th: 31
- 9th: 28
- 10th: ? → even → 31 + 3 = 34

Wait — positions:
1. 16 (odd)
2. 22 (even)
3. 19 (odd)
4. 25 (even)
5. 22 (odd)
6. 28 (even)
7. 25 (odd)
8. ? (even) → 31
9. ? (odd) → 28
10. ? (even) → 34

So:
- 8: 31
- 9: 28
- 10: 34

✔ Answer: 31, 28, 34

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7) 4, 12, 16, 48, 52, 156, 160, __, __, __

This is identical to #4!

Same numbers: 4, 12, 16, 48, 52, 156, 160...

So same pattern: ×3, +4, ×3, +4, ...

- 160 → ×3 = 480
- 480 → +4 = 484
- 484 → ×3 = 1452

✔ Answer: 480, 484, 1452

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8) 4, 8, 1, 2, -5, -10, -17, __, __, __

Try splitting into odd and even:

Odd positions (1st, 3rd, 5th, 7th):
- 4, 1, -5, -17

Differences:
- 4 → 1: -3
- 1 → -5: -6
- -5 → -17: -12

Differences: -3, -6, -12 → ×2 each time → next: -24

So: -17 -24 = -41

Even positions (2nd, 4th, 6th, 8th):
- 8, 2, -10, ?

Differences:
- 8 → 2: -6
- 2 → -10: -12
- -10 → ? → -24 → so -10 -24 = -34

So:
- 8th: -34
- 9th: -41
- 10th: ? → even → next even after -34 → difference pattern: -6, -12, -24 → next: -48 → -34 -48 = -82

So:
- 8: -34
- 9: -41
- 10: -82

✔ Answer: -34, -41, -82

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9) 22, 28, 21, 27, 20, 26, 19, __, __, __

Split into odd and even:

Odd positions (1st, 3rd, 5th, 7th):
- 22, 21, 20, 19 → decreasing by 1 → next: 18

Even positions (2nd, 4th, 6th, 8th):
- 28, 27, 26, ? → decreasing by 1 → next: 25

So:
- 8th: 25
- 9th: 18
- 10th: ? → even → 25 -1 = 24

So:
- 8: 25
- 9: 18
- 10: 24

✔ Answer: 25, 18, 24

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10) 1, 2, -4, -8, -14, -28, -34, __, __, __

Look at pattern:

- 1 → 2 (+1)
- 2 → -4 (-6)
- -4 → -8 (-4)
- -8 → -14 (-6)
- -14 → -28 (-14)
- -28 → -34 (-6)

Hmm — alternating?

Check differences:
+1, -6, -4, -6, -14, -6

Not clear.

Try grouping:

Maybe two interleaved sequences?

Odd positions (1st, 3rd, 5th, 7th):
- 1, -4, -14, -34

Differences:
- 1 → -4: -5
- -4 → -14: -10
- -14 → -34: -20

→ Differences: -5, -10, -20 → ×2 each time → next: -40 → -34 -40 = -74

Even positions (2nd, 4th, 6th, 8th):
- 2, -8, -28, ?

Differences:
- 2 → -8: -10
- -8 → -28: -20
- -28 → ? → -40 → -28 -40 = -68

So:
- 8th: -68
- 9th: -74
- 10th: ? → even → next even → -68 -80 = -148? Wait — pattern: -10, -20, -40 → ×2 → next: -80 → -68 -80 = -148

But let’s check:

Odd: 1, -4, -14, -34, -74 → diffs: -5, -10, -20, -40 → ×2 → yes

Even: 2, -8, -28, -68 → diffs: -10, -20, -40 → ×2 → next diff: -80 → -68 -80 = -148

So:
- 8: -68
- 9: -74
- 10: -148

✔ Answer: -68, -74, -148

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✔ Final Answers:

1) 92, 94, 188
2) -75, -79, -237
3) 13, 21, 34
4) 480, 484, 1452
5) 6, 1, 4
6) 31, 28, 34
7) 480, 484, 1452
8) -34, -41, -82
9) 25, 18, 24
10) -68, -74, -148

Let me know if you'd like explanations in a printable format!