Complete the Numerical Series worksheet for math practice.

Let’s solve each numerical series one by one, looking for patterns (arithmetic, geometric, alternating, recursive, etc.).

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

Pattern:
Look at the operations between terms:

- 4 → 8: ×2
- 8 → 10: +2
- 10 → 20: ×2
- 20 → 22: +2
- 22 → 44: ×2
- 44 → ? : +2 → 46
- 46 → ? : ×2 → 92
- 92 → ? : +2 → 94

✔ Answer: 46, 92, 94

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

This looks tricky. Let’s check differences or multiplicative patterns.

Group as pairs? Or look at positions:

Try recursive pattern:

From 1 to 3: ×3? But then 3 to -1? Not clear.

Check:

Term 1: 1
Term 2: 3 → 1×3 = 3
Term 3: -1 → 3 - 4 = -1?
Term 4: -3 → -1 × 3 = -3
Term 5: -7 → -3 - 4 = -7
Term 6: -21 → -7 × 3 = -21
→ So pattern: ×3, then -4, repeat.

So:

Term 7: -21 - 4 = -25
Term 8: -25 × 3 = -75
Term 9: -75 - 4 = -79

✔ Answer: -25, -75, -79

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

Classic Fibonacci sequence: each term is sum of two previous.

- 0+1=1
- 1+1=2
- 1+2=3
- 2+3=5
- 3+5=8
- 5+8=13
- 8+13=21
- 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

✔ Answer: 480, 484, 1452

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

Alternate pattern:

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

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

Next even position (8th): 8 - 2 = 6

Next odd position (9th): 3 - 2 = 1

Next even position (10th): 6 - 2 = 4

✔ Answer: 6, 1, 4

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

Pattern: Alternating sequences?

Positions:

1: 16
2: 22
3: 19
4: 25
5: 22
6: 28
7: 25

Odd positions (1,3,5,7): 16, 19, 22, 25 → +3 each time → next (9th): 28

Even positions (2,4,6,8): 22, 25, 28 → +3 → next (8th): 31, then (10th): 34

So:

Term 8 (even): 28 + 3 = 31
Term 9 (odd): 25 + 3 = 28
Term 10 (even): 31 + 3 = 34

✔ Answer: 31, 28, 34

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

Wait — this is identical to problem #4!

Same pattern: ×3, +4, ×3, +4...

We already solved it:

After 160: ×3 → 480, +4 → 484, ×3 → 1452

✔ Answer: 480, 484, 1452

*(Note: Probably a duplicate in the worksheet.)*

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

Look at differences:

4 → 8: +4
8 → 1: -7
1 → 2: +1
2 → -5: -7
-5 → -10: -5
-10 → -17: -7

Not obvious. Try grouping:

Maybe two interleaved sequences?

Odd positions: 4, 1, -5, -17 → differences: -3, -6, -12 → doubling subtracted?
-3, then -6 (×2), then -12 (×2) → next: -24 → -17 -24 = -41

Even positions: 8, 2, -10 → differences: -6, -12 → next: -24 → -10 -24 = -34

But we need next three terms: positions 8,9,10.

Position 8 (even): next even term after -10 → -10 -24 = -34
Position 9 (odd): after -17 → -17 -24 = -41
Position 10 (even): after -34 → -34 -48? Wait, pattern was -6, -12, -24 → next difference -48?
→ -34 -48 = -82

Wait — let’s verify:

Odd indices (1,3,5,7,9): 4, 1, -5, -17, ?

Differences: 1-4=-3; -5-1=-6; -17-(-5)=-12 → so -3, -6, -12 → multiply by 2 each time → next diff = -24 → -17 + (-24) = -41

Even indices (2,4,6,8,10): 8, 2, -10, ?, ?

Differences: 2-8=-6; -10-2=-12 → next diff = -24 → -10 + (-24) = -34
Then next diff = -48 → -34 + (-48) = -82

So:

Term 8 (even): -34
Term 9 (odd): -41
Term 10 (even): -82

✔ Answer: -34, -41, -82

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

Pattern: Alternating?

Odd positions: 22, 21, 20, 19 → decreasing by 1 → next (9th): 18

Even positions: 28, 27, 26 → decreasing by 1 → next (8th): 25, then (10th): 24

So:

Term 8 (even): 26 -1 = 25
Term 9 (odd): 19 -1 = 18
Term 10 (even): 25 -1 = 24

✔ Answer: 25, 18, 24

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

Look at operations:

1 → 2: ×2
2 → -4: -6?
-4 → -8: ×2
-8 → -14: -6
-14 → -28: ×2
-28 → -34: -6
→ Pattern: ×2, then -6, repeat.

So:

-34 → ×2 = -68
-68 → -6 = -74
-74 → ×2 = -148

✔ Answer: -68, -74, -148

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## ✔ Final Answers:

1) 46, 92, 94
2) -25, -75, -79
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 visual chart or explanation for any specific one!