Math worksheet for completing numerical series with ten problems.

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, ?

Odd positions:
- 1 → -1 (-2)
- -1 → -7 (-6)
- -7 → -25 (-18)

Differences: -2, -6, -18 → multiply by 3 each time?
-2 × 3 = -6, -6 × 3 = -18 → next would be -54

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

But we need to fill after -25 (which is position 7), so next is position 8 (even), then 9 (odd), etc.

We already have:
Positions:
1: 1
2: 3
3: -1
4: -3
5: -7
6: -21
7: -25
8: ?
9: ?
10: ?

Now check even positions: 3, -3, -21, ?

- 3 → -3 (-6)
- -3 → -21 (-18)
- -21 → ? (-54?) since -6 × 3 = -18, -18 × 3 = -54

So: -21 - 54 = -75

So even position 8 = -75

Now odd position 9: from earlier, -25 + (-54) = -79

Then even position 10: next in even sequence: previous difference was -54, next would be -54 × 3 = -162

So: -75 - 162 = -237

Wait — but we only need three more terms: after -25, so:

- 8th term (even): -75
- 9th term (odd): -79
- 10th term (even): -237

But let’s double-check the odd sequence:

Odd indices: pos 1,3,5,7,9,...

- 1 → -1 (diff -2)
- -1 → -7 (diff -6)
- -7 → -25 (diff -18)
- -25 → ? (diff -54) → -79 ✔

Even indices: pos 2,4,6,8,10,...

- 3 → -3 (diff -6)
- -3 → -21 (diff -18)
- -21 → ? (diff -54) → -75
- -75 → ? (diff -162) → -237

Yes.

✔ 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,
- 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 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

Pattern: ×3, +4, ×3, +4, ...

✔ Answer: 480, 484, 1452

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

Split into two alternating sequences:

Odd positions: 9, 7, 5, 3, ... → decreasing by 2 → next: 1

Even positions: 12, 10, 8, ?, → decreasing by 2 → next: 6

So:
- 8th term (even): 6
- 9th term (odd): 1
- 10th term (even): 4

Sequence:
1: 9
2: 12
3: 7
4: 10
5: 5
6: 8
7: 3
8: ? → 6
9: ? → 1
10: ? → 4

✔ Answer: 6, 1, 4

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

Look at positions:

Odd: 16, 19, 22, 25 → increasing by 3 → next: 28
Even: 22, 25, 28, ? → increasing by 3 → next: 31

So:
- 8th term (even): 31
- 9th term (odd): 28
- 10th term (even): 34

Sequence:
1: 16
2: 22
3: 19
4: 25
5: 22
6: 28
7: 25
8: ? → 31
9: ? → 28
10: ? → 34

Wait: odd positions: 16, 19, 22, 25, 28 → yes, +3 each
Even: 22, 25, 28, 31, 34 → +3 each

✔ Answer: 31, 28, 34

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

Wait — this is identical to #4!

So same as #4: ×3, +4, ×3, +4...

After 160:
- ×3 = 480
- +4 = 484
- ×3 = 1452

✔ Answer: 480, 484, 1452

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

Look at pattern.

Group as pairs?

4, 8 → both positive
1, 2 → smaller
-5, -10 → negative, doubling
-17, ? → maybe?

Or try alternating sequences.

Odd positions: 4, 1, -5, -17, ?
- 4 → 1 (-3)
- 1 → -5 (-6)
- -5 → -17 (-12)
- Differences: -3, -6, -12 → ×2 each time → next: -24
→ -17 - 24 = -41

Even positions: 8, 2, -10, ?, ?
- 8 → 2 (-6)
- 2 → -10 (-12)
- -10 → ? (-24) → -34
- Then next: -34 → ? (-48) → -82

So:
- 8th term (even): -34
- 9th term (odd): -41
- 10th term (even): -82

✔ Answer: -34, -41, -82

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

Odd positions: 22, 21, 20, 19 → decreasing by 1 → next: 18
Even positions: 28, 27, 26, ? → decreasing by 1 → next: 25

So:
- 8th term (even): 25
- 9th term (odd): 18
- 10th term (even): 24

Sequence:
1: 22
2: 28
3: 21
4: 27
5: 20
6: 26
7: 19
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.

Try grouping:

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

But not consistent.

Try alternating sequences.

Odd positions: 1, -4, -14, -34, ?
- 1 → -4 (-5)
- -4 → -14 (-10)
- -14 → -34 (-20)
- Differences: -5, -10, -20 → ×2 each → next: -40
→ -34 - 40 = -74

Even positions: 2, -8, -28, ?
- 2 → -8 (-10)
- -8 → -28 (-20)
- -28 → ? (-40) → -68

So:
- 8th term (even): -68
- 9th term (odd): -74
- 10th term (even): -136? Wait, next difference would be -80?

Wait: differences in even: -10, -20, -40 → ×2 each → next: -80

So: -68 - 80 = -148

But we need:
- 8th: even → -68
- 9th: odd → -74
- 10th: even → -148

Wait, no: after -68, next even term is 10th, so:

- Even: 2, -8, -28, -68, -148 → differences: -10, -20, -40, -80 → yes

Odd: 1, -4, -14, -34, -74 → diff: -5, -10, -20, -40 → yes

✔ 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 a printed version or explanation for any specific one!