Complete the Numerical Series worksheet with ten sequences to solve.

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 pairs:

(1, 3), (-1, -3), (-7, -21), (-25, ?)

From first pair: 1 → 3 (×3? No, +2)

Wait — another idea: Maybe every third number is related?

Try looking at indices:

| n | value |
|---|-------|
| 1 | 1 |
| 2 | 3 |
| 3 | -1 |
| 4 | -3 |
| 5 | -7 |
| 6 | -21 |
| 7 | -25 |
| 8 | ? |
| 9 | ? |
|10 | ? |

Now consider odd and even separately.

Odd positions (1,3,5,7): 1, -1, -7, -25
Differences:
- -1 - 1 = -2
- -7 - (-1) = -6
- -25 - (-7) = -18

So: -2, -6, -18 → multiplying by 3 each time!

→ Next difference: -18 × 3 = -54
→ Next term: -25 - 54 = -79

So 9th term (odd position) = -79

Now even positions (2,4,6,8): 3, -3, -21, ?

Differences:
- -3 - 3 = -6
- -21 - (-3) = -18

→ -6, -18 → ×3 → next difference: -54
→ Next term: -21 - 54 = -75

So 8th term = -75

Now 10th term (even) would be next in even sequence:
-75 → ? (next difference?)
We had: -6, -18, -54 → ×3 again → next: -162
→ -75 - 162 = -237

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

So:

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

But let’s verify consistency.

Wait — perhaps better: The pattern in even positions:

3 → -3 (× -1)
-3 → -21 (×7)? Not consistent.

Alternative idea: Look at recurrence.

Wait — from earlier:

Odd positions: 1, -1, -7, -25, ?

Differences: -2, -6, -18 → ×3 → next: -54 → -25 -54 = -79 ✔

Even positions: 3, -3, -21, ?

3 → -3: ×(-1)
-3 → -21: ×7 → not helpful.

But:
3, -3, -21, ?

Maybe:
3 → -3: subtract 6
-3 → -21: subtract 18
Then subtract 54? → -21 -54 = -75 → same as before.

Yes! So pattern: subtracting 6, 18, 54... which is ×3 each time.

So:
-21 → -75 (subtract 54)
Then next: subtract 54×3 = 162 → -75 -162 = -237

So even terms: 3, -3, -21, -75, -237,...

Thus:

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

✔ Answer: -75, -79, -237

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

This is clearly the Fibonacci sequence:

Each term is sum of previous two.

- 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 analyze:

- 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 after 160:

- ×3 = 480
- +4 = 484
- ×3 = 1452

✔ Answer: 480, 484, 1452

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

Look at alternating subsequences.

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

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

So:

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

Sequence:
- Position 7: 3 (odd)
- Position 8: ? (even) → 6
- Position 9: ? (odd) → 1
- Position 10: ? (even) → 4

✔ Answer: 6, 1, 4

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

Look at pattern:

Group: (16,22), (19,25), (22,28), (25,?)

First numbers: 16, 19, 22, 25 → +3 each

Second numbers: 22, 25, 28, ? → +3 each

So next pair: (25, 31)

Then next: (28, 34), etc.

But let's list:

- 16, 22 → +6
- 22 → 19 → -3
- 19 → 25 → +6
- 25 → 22 → -3
- 22 → 28 → +6
- 28 → 25 → -3
- 25 → ? → +6 → 31
- 31 → ? → -3 → 28
- 28 → ? → +6 → 34

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

✔ Answer: 31, 28, 34

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

Wait — this is identical to #4!

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

Same as problem 4.

So answer is same: 480, 484, 1452

✔ Answer: 480, 484, 1452

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

Let’s break into two sequences?

Odd positions: 4, 1, -5, -17, ?
Even positions: 8, 2, -10, ?

Odd: 4, 1, -5, -17

Differences:
- 1 - 4 = -3
- -5 - 1 = -6
- -17 - (-5) = -12

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

→ Next term: -17 -24 = -41

Even: 8, 2, -10, ?

Differences:
- 2 - 8 = -6
- -10 - 2 = -12 → ×2 → next: -24

→ Next: -10 -24 = -34

Now, continue:

Next odd: after -41 → difference: -48 → -41 -48 = -89

But we need three more terms: 8th, 9th, 10th

Current:

- 7th term: -17 (odd)
- 8th term: ? (even) → -34
- 9th term: ? (odd) → -41
- 10th term: ? (even) → next even: -34 -48 = -82? Wait.

Even sequence: 8, 2, -10, ?, ?

Differences: -6, -12, → ×2 → next: -24 → -10 -24 = -34
Then: -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, __, __, __

Alternating pattern?

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

Even positions: 28, 27, 26, ?

So even: 28, 27, 26 → next: 25

Then next odd: 19 -1 = 18

Then even: 25 -1 = 24

So:

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

✔ Answer: 25, 18, 24

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

Let’s examine:

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

Look at changes:

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

Not clear.

Try splitting into two sequences?

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

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

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

Differences:
- -4 - 1 = -5
- -14 - (-4) = -10
- -34 - (-14) = -20

→ -5, -10, -20 → ×2 → next: -40 → next term: -34 -40 = -74

Even: 2, -8, -28, ?

Differences:
- -8 - 2 = -10
- -28 - (-8) = -20 → ×2 → next: -40 → -28 -40 = -68

Now:
- 8th term: -68 (even)
- 9th term: -74 (odd)
- 10th term: ? (even) → next even: -68 -80 = -148? (since -40 ×2 = -80)

Wait: pattern in even differences: -10, -20, -40 → ×2 → next: -80 → -68 -80 = -148

So:

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

But let's confirm:

Sequence:
1 (1), 2 (2), -4 (3), -8 (4), -14 (5), -28 (6), -34 (7), ?, ?, ?

So:
- 8th: even → -68
- 9th: odd → -74
- 10th: even → -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

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