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, this seems messy. Let's look at groups:

Try splitting into two interleaved sequences?

Odd positions: 1, -1, -7, -25
Even positions: 3, -3, -21, ?

But maybe not.

Alternatively, check if it's a recursive pattern.

Another idea: Look at the changes:

- 1 to 3: +2
- 3 to -1: -4
- -1 to -3: -2
- -3 to -7: -4
- -7 to -21: -14
- -21 to -25: -4

Hmm, alternating between -4 and other values?

Wait:
+2, -4, -2, -4, -14, -4

No clear pattern.

Wait — let's try grouping:

Look at:
- 1, 3 → then -1, -3 → then -7, -21 → then -25,...

Wait:
After -21, next is -25 → difference of -4

But before that: -7 to -21 = ×3? No, -7×3 = -21 → yes!

Then -21 → -25 = -4

Now go back:

-3 → -7 = -4? No, -3 to -7 is -4? Yes: -3 - 4 = -7

Wait, let’s list with indices:

| n | value |
|---|-------|
| 1 | 1 |
| 2 | 3 |
| 3 | -1 |
| 4 | -3 |
| 5 | -7 |
| 6 | -21 |
| 7 | -25 |

Now look at odd and even positions separately.

Odd positions (n=1,3,5,7): 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 → so next would be -18 × 3 = -54

So next odd term: -25 + (-54) = -79

So 9th term = -79

Now even positions (n=2,4,6): 3, -3, -21

- 3 → -3: -6
- -3 → -21: -18

-6 → -18 → ×3 again?

Next difference: -18 × 3 = -54

So next even term: -21 - 54 = -75

So 8th term = -75

Now we need 8th, 9th, 10th terms.

We have:

- 7th: -25
- 8th: ? → even position → -75
- 9th: ? → odd position → -79
- 10th: ? → even → next after -75

Even sequence: 3, -3, -21, -75

Differences: -6, -18, -54 → ×3 each time

Next: -54 × 3 = -162 → next even term: -75 - 162 = -237

So:

- 8th: -75
- 9th: -79
- 10th: -237

✔ 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 previous:

- 0, 1, 1, 2, 3, 5, 8, ...
- 8 + 5 = 13
- 13 + 8 = 21
- 21 + 13 = 34

✔ Answer: 13, 21, 34

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

Let’s look at the 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

So pattern: ×3, +4, ×3, +4, ×3, +4, ...

✔ Answer: 480, 484, 1452

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

Group in pairs?

Or look at alternating sequences.

Let’s separate odd and even positions:

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

So next odd: 1 (after 3), then -1, etc.

But we need 8th, 9th, 10th → even positions are 2nd, 4th, 6th, 8th, 10th

Even positions: 12, 10, 8, ?, ?

→ 12, 10, 8 → decreasing by 2 → next: 6, 4

So:

- 8th term (even): 6
- 10th term (even): 4
- 9th term (odd): after 3 → 1

So:

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

✔ Answer: 6, 1, 4

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

Look at positions:

1: 16
2: 22
3: 19
4: 25
5: 22
6: 28
7: 25
8: ?
9: ?
10: ?

Now group:

Odd positions: 16, 19, 22, 25 → increasing by 3

- 16 → 19 (+3)
- 19 → 22 (+3)
- 22 → 25 (+3)
- Next: 25 + 3 = 28 → 9th term

Even positions: 22, 25, 28 → increasing by 3

- 22 → 25 (+3)
- 25 → 28 (+3)
- Next: 28 + 3 = 31 → 8th term

Then 10th term: next even → 31 + 3 = 34

So:

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

✔ Answer: 31, 28, 34

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

This is identical to #4!

Yes: 4, 12, 16, 48, 52, 156, 160, ...

Same as #4 → ×3, +4, ×3, +4, ...

So same answer:

- 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, __, __, __

Let’s see:

4 → 8 → ×2
8 → 1 → /8? No.
1 → 2 → ×2
2 → -5 → -7?
-5 → -10 → ×2
-10 → -17 → -7

Hmm.

Try splitting into two sequences?

Odd positions: 4, 1, -5, -17

Even positions: 8, 2, -10, ?

Odd: 4, 1, -5, -17

- 4 → 1: -3
- 1 → -5: -6
- -5 → -17: -12 → differences: -3, -6, -12 → ×2 each time

Next difference: -24 → -17 -24 = -41

So 9th term = -41

Even: 8, 2, -10, ?

- 8 → 2: -6
- 2 → -10: -12 → ×2

Next: -24 → -10 -24 = -34

So 8th term = -34

Now 10th term: next even → next difference: -48 → -34 -48 = -82

So:

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

✔ Answer: -34, -41, -82

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

Look at pattern:

22, 28 → +6
21, 27 → +6
20, 26 → +6
19, ? → +6 → 25?

But let's see the full sequence:

Positions:

1: 22
2: 28
3: 21
4: 27
5: 20
6: 26
7: 19
8: ?
9: ?
10: ?

Odd positions: 22, 21, 20, 19 → decreasing by 1

So next odd: 18 (9th), then 17 (11th)

Even positions: 28, 27, 26, ? → decreasing by 1

So 8th: 25, 10th: 24

So:

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

✔ Answer: 25, 18, 24

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

Let’s analyze:

1, 2, -4, -8, -14, -28, -34

Check differences:

- 1 → 2: +1
- 2 → -4: -6
- -4 → -8: -4
- -8 → -14: -6
- -14 → -28: -14
- -28 → -34: -6

Not consistent.

Wait: look at signs and magnitude.

Try grouping:

1, 2 → positive
-4, -8 → negative
-14, -28 → negative? But -34 is also negative.

Wait: perhaps alternating pattern?

Look at:

1, 2 → +1, +2
Then -4, -8 → ×2
Then -14, -28 → ×2
Then -34 → next?

Wait: -14 → -28 (×2), but -28 → -34 → +(-6)

Hmm.

Alternative: split into two sequences?

Odd positions: 1, -4, -14, -34

Even positions: 2, -8, -28, ?

Odd: 1, -4, -14, -34

- 1 → -4: -5
- -4 → -14: -10
- -14 → -34: -20 → differences: -5, -10, -20 → ×2 each

Next: -40 → -34 -40 = -74 → 9th term

Even: 2, -8, -28, ?

- 2 → -8: -10
- -8 → -28: -20 → ×2

Next: -40 → -28 -40 = -68 → 8th term

Then 10th term: next even → -68 -80 = -148

So:

- 8th: -68
- 9th: -74
- 10th: -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 simpler language or visual patterns!