Number Patterns worksheet for students to identify and extend sequences.

Let's solve each of the number pattern problems step by step, identify the pattern rule, and then extend the sequence with the next two numbers.

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1) 4, 8, 11, 22, 25, 50, 53, ___ , ___

Step-by-step analysis:
- 4 → 8: multiply by 2
- 8 → 11: add 3
- 11 → 22: multiply by 2
- 22 → 25: add 3
- 25 → 50: multiply by 2
- 50 → 53: add 3

Pattern: Alternate between multiplying by 2 and adding 3.

So:
- 53 × 2 = 106
- 106 + 3 = 109

✔ Next two numbers: 106, 109
✔ Pattern Rule: Multiply by 2, then add 3 (alternating)

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2) 20, 15, 25, 20, 30, 25, 35, ___ , ___

Look at the sequence:
- 20, 15, 25, 20, 30, 25, 35,...

Break it into two interleaved sequences:

- Odd positions (1st, 3rd, 5th, 7th): 20, 25, 30, 35 → increasing by 5
- Even positions (2nd, 4th, 6th): 15, 20, 25 → increasing by 5

So:
- Next odd position (9th): 35 + 5 = 40
- Next even position (8th): 25 + 5 = 30

Wait — we need to fill in the 8th and 9th terms. The last given is 35 (7th term), so:

- 8th term: next in even sequence → 25 + 5 = 30
- 9th term: next in odd sequence → 35 + 5 = 40

✔ Next two numbers: 30, 40
✔ Pattern Rule: Two alternating sequences: one increases by 5 starting at 20 (odd positions), the other increases by 5 starting at 15 (even positions).

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3) 48, 57, 65, 72, 78, 83, 87, ___ , ___

Check differences:
- 57 - 48 = 9
- 65 - 57 = 8
- 72 - 65 = 7
- 78 - 72 = 6
- 83 - 78 = 5
- 87 - 83 = 4

Differences: 9, 8, 7, 6, 5, 4 → decreasing by 1 each time.

So next difference: 3 → 87 + 3 = 90
Then: 2 → 90 + 2 = 92

✔ Next two numbers: 90, 92
✔ Pattern Rule: Subtract 1 from the previous difference each time; start with +9, then +8, etc.

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4) 2, 4, 6, 12, 14, 28, 30, ___ , ___

Analyze:
- 2 → 4: +2
- 4 → 6: +2
- 6 → 12: ×2
- 12 → 14: +2
- 14 → 28: ×2
- 28 → 30: +2

So pattern: +2, +2, ×2, repeat?

Let’s see:
- 2 → 4 (+2)
- 4 → 6 (+2)
- 6 → 12 (×2)
- 12 → 14 (+2)
- 14 → 28 (×2)
- 28 → 30 (+2)

So the cycle is: +2, +2, ×2, repeating?

But after 30, what comes?

From 30:
- Should be ×2 → 30 × 2 = 60
- Then +2 → 62
- Then +2 → 64?

Wait — but let's check the pattern:

Actually, the operations are:
- +2, +2, ×2, +2, ×2, +2 → not consistent.

Wait — better to look at groups:

Group: (2,4,6), (12,14,28), (30,...)

Wait — maybe alternate:

From 6 → 12: ×2
Then 12 → 14: +2
14 → 28: ×2
28 → 30: +2
So likely: ×2, +2, ×2, +2, ...

But before that: 2 → 4 (+2), 4 → 6 (+2)

So maybe the pattern is:
- Start: +2, +2
- Then alternate ×2 and +2?

But let's reindex:

Sequence: 2, 4, 6, 12, 14, 28, 30, ?, ?

Try grouping:
- 2, 4, 6 → +2, +2
- 6 → 12 → ×2
- 12 → 14 → +2
- 14 → 28 → ×2
- 28 → 30 → +2

So after 30:
- ×2 → 60
- +2 → 62

But wait — the ×2 and +2 alternate after the first two steps.

So:
- 30 → ×2 = 60
- 60 → +2 = 62

But then what? Maybe continue: ×2 → 124? But that seems off.

Alternatively, maybe it's:
- Every third term is ×2?

But 6 → 12 (×2), 14 → 28 (×2), 30 → ? → ×2 = 60

Then 60 → ? → +2 = 62

Yes — matches.

So pattern: after the first two additions, alternate ×2 and +2.

But actually, the pattern seems to be:
- +2, +2, ×2, +2, ×2, +2, ×2, +2, ...

So:
- 30 → ×2 = 60
- 60 → +2 = 62

✔ Next two numbers: 60, 62
✔ Pattern Rule: After the first two numbers, alternate between multiplying by 2 and adding 2, starting with ×2.

Wait — but the sequence goes: 6 → 12 (×2), 12 → 14 (+2), 14 → 28 (×2), 28 → 30 (+2), 30 → ? → ×2 = 60, 60 → ? → +2 = 62

So yes, alternating ×2 and +2 from position 4 onward.

But the very beginning was +2, +2 — maybe a different start.

But for continuation: from 30, apply ×2 → 60, then +2 → 62.

✔ So: 60, 62

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5) 3, 6, 5, 10, 9, 18, 17, ___ , ___

Look at:
- 3 → 6: ×2
- 6 → 5: -1
- 5 → 10: ×2
- 10 → 9: -1
- 9 → 18: ×2
- 18 → 17: -1
- 17 → ?

So pattern: ×2, -1, ×2, -1, ×2, -1, ...

So after 17:
- ×2 → 34
- -1 → 33

✔ Next two numbers: 34, 33
✔ Pattern Rule: Multiply by 2, then subtract 1, alternating.

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6) 53, 50, 47, 43, 40, 35, 32, ___ , ___

Check differences:
- 53 → 50: -3
- 50 → 47: -3
- 47 → 43: -4
- 43 → 40: -3
- 40 → 35: -5
- 35 → 32: -3

Hmm — not obvious.

List:
53, 50, 47, 43, 40, 35, 32

Differences:
-3, -3, -4, -3, -5, -3

So every third difference is decreasing: -4, -5, next might be -6?

But positions:
- 1→2: -3
- 2→3: -3
- 3→4: -4
- 4→5: -3
- 5→6: -5
- 6→7: -3

So pattern: -3, -3, -4, -3, -5, -3, ...

So the non-3 differences are at positions 3, 5, 7: -4, -5, ? → probably -6

So:
- 32 → ? : -6 → 26
- 26 → ? : -3 → 23

Wait — but let’s check if the pattern is: every third term decreases by an increasing amount?

Sequence:
- 53, 50, 47 → -3, -3
- 47 → 43 → -4
- 43 → 40 → -3
- 40 → 35 → -5
- 35 → 32 → -3

So the large drops are at odd positions (after the first two):
- 47 → 43 (-4)
- 40 → 35 (-5)
- 32 → ? → (-6)

And in between: -3

So:
- 32 → -6 → 26
- 26 → -3 → 23

✔ Next two numbers: 26, 23
✔ Pattern Rule: Alternating between subtracting 3 and subtracting increasing values: -3, -3, -4, -3, -5, -3, -6, -3...

Or: subtract 3, except every third step (starting from 3rd) subtracts 4, 5, 6,...

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7) 94, 91, 86, 79, 70, 59, 46, ___ , ___

Check differences:
- 94 → 91: -3
- 91 → 86: -5
- 86 → 79: -7
- 79 → 70: -9
- 70 → 59: -11
- 59 → 46: -13

Differences: -3, -5, -7, -9, -11, -13 → decreasing by 2 each time

So next: -15 → 46 - 15 = 31
Then: -17 → 31 - 17 = 14

✔ Next two numbers: 31, 14
✔ Pattern Rule: Subtract consecutive odd numbers: -3, -5, -7, -9, -11, -13, -15, -17,...

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8) 63, 67, 64, 68, 65, 69, 66, ___ , ___

Look at:
- 63 → 67: +4
- 67 → 64: -3
- 64 → 68: +4
- 68 → 65: -3
- 65 → 69: +4
- 69 → 66: -3
- 66 → ?

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

So after 66:
- +4 → 70
- -3 → 67

✔ Next two numbers: 70, 67
✔ Pattern Rule: Alternate between adding 4 and subtracting 3.

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9) 19, 25, 32, 40, 49, 59, 70, ___ , ___

Check differences:
- 19 → 25: +6
- 25 → 32: +7
- 32 → 40: +8
- 40 → 49: +9
- 49 → 59: +10
- 59 → 70: +11

So differences: +6, +7, +8, +9, +10, +11 → increasing by 1

So next: +12 → 70 + 12 = 82
Then: +13 → 82 + 13 = 95

✔ Next two numbers: 82, 95
✔ Pattern Rule: Add consecutive integers starting from 6.

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10) 29, 31, 34, 38, 43, 49, 56, ___ , ___

Differences:
- 29 → 31: +2
- 31 → 34: +3
- 34 → 38: +4
- 38 → 43: +5
- 43 → 49: +6
- 49 → 56: +7

So differences: +2, +3, +4, +5, +6, +7 → increasing by 1

Next: +8 → 56 + 8 = 64
Then: +9 → 64 + 9 = 73

✔ Next two numbers: 64, 73
✔ Pattern Rule: Add consecutive integers starting from 2.

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## ✔ Final Answers:

| # | Next Two Numbers | Pattern Rule |
|---|------------------|-------------|
| 1) | 106, 109 | Multiply by 2, then add 3 (alternate) |
| 2) | 30, 40 | Two alternating sequences: odd positions increase by 5 (20,25,30,35,40), even positions increase by 5 (15,20,25,30) |
| 3) | 90, 92 | Differences decrease by 1: +9, +8, +7, +6, +5, +4, +3, +2 |
| 4) | 60, 62 | After initial +2,+2, alternate ×2 and +2 |
| 5) | 34, 33 | ×2, then -1, alternate |
| 6) | 26, 23 | Subtract 3, except every third step: subtract 4,5,6,... |
| 7) | 31, 14 | Subtract consecutive odd numbers: -3,-5,-7,-9,-11,-13,-15,-17 |
| 8) | 70, 67 | +4, -3, alternate |
| 9) | 82, 95 | Add consecutive integers starting from 6: +6,+7,+8,... |
|10) | 64, 73 | Add consecutive integers starting from 2: +2,+3,+4,... |

Let me know if you'd like this formatted as a printable answer sheet!