Number Patterns & Sequences worksheet - Free Printable
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Step-by-step solution for: Number Patterns & Sequences worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Number Patterns & Sequences worksheet
Let’s solve each sequence step by step. We’ll look for the pattern in each one — whether it’s adding, multiplying, subtracting, or alternating operations.
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EXERCISE 1
(a) 2, 4, 8, 16, ___, ___, ___
Look:
2 × 2 = 4
4 × 2 = 8
8 × 2 = 16
So next:
16 × 2 = 32
32 × 2 = 64
64 × 2 = 128
→ Answer: 32, 64, 128
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(b) 1, 4, 7, 10, ___, ___, ___
Look:
1 + 3 = 4
4 + 3 = 7
7 + 3 = 10
So add 3 each time:
10 + 3 = 13
13 + 3 = 16
16 + 3 = 19
→ Answer: 13, 16, 19
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(c) 7, 14, 21, 28, ___, ___, ___
Look:
7 × 1 = 7
7 × 2 = 14
7 × 3 = 21
7 × 4 = 28
So multiply 7 by 5, 6, 7:
7 × 5 = 35
7 × 6 = 42
7 × 7 = 49
→ Answer: 35, 42, 49
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(d) 2, 20, 200, 2000, ___, ___, ___
Look:
2 × 10 = 20
20 × 10 = 200
200 × 10 = 2000
So multiply by 10 each time:
2000 × 10 = 20,000
20,000 × 10 = 200,000
200,000 × 10 = 2,000,000
→ Answer: 20000, 200000, 2000000
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(e) 6, 12, 18, 24, ___, ___, ___
Look:
6 × 1 = 6
6 × 2 = 12
6 × 3 = 18
6 × 4 = 24
So multiply 6 by 5, 6, 7:
6 × 5 = 30
6 × 6 = 36
6 × 7 = 42
→ Answer: 30, 36, 42
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EXERCISE 2
(a) 58, 68, 63, 73, 68, ___, ___, ___
Look at changes:
58 → 68 (+10)
68 → 63 (-5)
63 → 73 (+10)
73 → 68 (-5)
Pattern: +10, -5, +10, -5... so next is +10, then -5, then +10
68 + 10 = 78
78 - 5 = 73
73 + 10 = 83
→ Answer: 78, 73, 83
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(b) 7, 8, 15, 23, 38, ___, ___, ___
Look:
7 + 8 = 15
8 + 15 = 23
15 + 23 = 38
This is Fibonacci-style: each number is sum of previous two.
Next:
23 + 38 = 61
38 + 61 = 99
61 + 99 = 160
→ Answer: 61, 99, 160
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(c) 55, 34, 21, 13, 8, ___, ___, ___
Look:
55 - 34 = 21
34 - 21 = 13
21 - 13 = 8
So subtract previous from current to get next? Wait — actually:
It’s going down:
55 → 34 (subtract 21)
34 → 21 (subtract 13)
21 → 13 (subtract 8)
13 → 8 (subtract 5)
Wait — let’s check differences between terms:
Actually, this looks like reverse Fibonacci:
55, 34, 21, 13, 8 — these are Fibonacci numbers backwards!
Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...
So after 8 comes 5, then 3, then 2.
Check:
8 - 5 = 3? No — we’re listing them descending.
Sequence: 55, 34, 21, 13, 8 → next should be 5, then 3, then 2.
Because 8 = 5 + 3, but here we’re just continuing the Fibonacci sequence backward.
Yes: 8 → 5 → 3 → 2
→ Answer: 5, 3, 2
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(d) 576, 288, 144, 72, ___, ___, ___
Look:
576 ÷ 2 = 288
288 ÷ 2 = 144
144 ÷ 2 = 72
So divide by 2 each time:
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
→ Answer: 36, 18, 9
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(e) 47, 40, 45, 38, 43, 36, 41, ___, ___, ___
Look at pairs or alternates:
Odd positions: 47, 45, 43, 41 → decreasing by 2
Even positions: 40, 38, 36 → also decreasing by 2
So next term is position 8 (even): 36 - 2 = 34
Then position 9 (odd): 41 - 2 = 39
Then position 10 (even): 34 - 2 = 32
Wait — let’s list with indices:
1: 47
2: 40
3: 45
4: 38
5: 43
6: 36
7: 41
8: ? → even → 36 - 2 = 34
9: ? → odd → 41 - 2 = 39
10: ? → even → 34 - 2 = 32
But wait — let’s verify the pattern again:
From 47 to 40: -7
40 to 45: +5
45 to 38: -7
38 to 43: +5
43 to 36: -7
36 to 41: +5
So pattern: -7, +5, -7, +5, -7, +5...
So next: 41 - 7 = 34
Then 34 + 5 = 39
Then 39 - 7 = 32
Same result.
→ Answer: 34, 39, 32
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EXERCISE 3
(a) 2, 3, 5, 7, 11, ___, ___
These are prime numbers!
Primes: 2, 3, 5, 7, 11, next is 13, then 17
→ Answer: 13, 17
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(b) 21, 22, 24, 25, 26, ___, ___
Look:
21 → 22 (+1)
22 → 24 (+2)
24 → 25 (+1)
25 → 26 (+1) — wait, that breaks pattern?
Wait — maybe skip some numbers?
List: 21, 22, 24, 25, 26
Differences: +1, +2, +1, +1 — not clear.
Alternative: Maybe it’s all numbers except multiples of 3?
21 is multiple of 3 — included? Hmm.
Wait — perhaps it’s increasing by 1, then 2, then 1, then 1 — doesn’t make sense.
Another idea: Maybe it’s consecutive integers skipping nothing? But 23 is missing.
21, 22, [skip 23], 24, 25, 26 — why skip 23?
Wait — 23 is prime, others aren’t? 21 no, 22 no, 24 no, 25 no, 26 no — 23 is prime, so maybe they skipped primes? But 2 and 3 were in part (a).
Wait — look again: 21, 22, 24, 25, 26 — what’s missing? 23.
Perhaps it’s numbers that are NOT prime? But 21=3×7, 22=2×11, 24=many, 25=5², 26=2×13 — all composite. 23 is prime — so maybe they’re listing composite numbers starting from 21.
Composite numbers after 20: 21, 22, 24, 25, 26, 27, 28...
So next: 27, 28
→ Answer: 27, 28
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(c) 1, 4, 9, 16, 25, ___, ___
These are perfect squares:
1²=1, 2²=4, 3²=9, 4²=16, 5²=25
Next: 6²=36, 7²=49
→ Answer: 36, 49
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(d) 47, 49, 51, 53, 55, 57, ___, ___
All odd numbers increasing by 2:
47, 49, 51, 53, 55, 57 → next 59, 61
→ Answer: 59, 61
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(e) 1, 8, 27, 64, ___, ___
Cubes:
1³=1, 2³=8, 3³=27, 4³=64
Next: 5³=125, 6³=216
→ Answer: 125, 216
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(f) 53, 59, 61, 67, ___, ___
Prime numbers!
After 53: 59 (next prime), then 61, then 67 — yes, all primes.
Next primes after 67: 71, 73
→ Answer: 71, 73
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Final Answer:
EXERCISE 1
(a) 32, 64, 128
(b) 13, 16, 19
(c) 35, 42, 49
(d) 20000, 200000, 2000000
(e) 30, 36, 42
EXERCISE 2
(a) 78, 73, 83
(b) 61, 99, 160
(c) 5, 3, 2
(d) 36, 18, 9
(e) 34, 39, 32
EXERCISE 3
(a) 13, 17
(b) 27, 28
(c) 36, 49
(d) 59, 61
(e) 125, 216
(f) 71, 73
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EXERCISE 1
(a) 2, 4, 8, 16, ___, ___, ___
Look:
2 × 2 = 4
4 × 2 = 8
8 × 2 = 16
So next:
16 × 2 = 32
32 × 2 = 64
64 × 2 = 128
→ Answer: 32, 64, 128
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(b) 1, 4, 7, 10, ___, ___, ___
Look:
1 + 3 = 4
4 + 3 = 7
7 + 3 = 10
So add 3 each time:
10 + 3 = 13
13 + 3 = 16
16 + 3 = 19
→ Answer: 13, 16, 19
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(c) 7, 14, 21, 28, ___, ___, ___
Look:
7 × 1 = 7
7 × 2 = 14
7 × 3 = 21
7 × 4 = 28
So multiply 7 by 5, 6, 7:
7 × 5 = 35
7 × 6 = 42
7 × 7 = 49
→ Answer: 35, 42, 49
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(d) 2, 20, 200, 2000, ___, ___, ___
Look:
2 × 10 = 20
20 × 10 = 200
200 × 10 = 2000
So multiply by 10 each time:
2000 × 10 = 20,000
20,000 × 10 = 200,000
200,000 × 10 = 2,000,000
→ Answer: 20000, 200000, 2000000
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(e) 6, 12, 18, 24, ___, ___, ___
Look:
6 × 1 = 6
6 × 2 = 12
6 × 3 = 18
6 × 4 = 24
So multiply 6 by 5, 6, 7:
6 × 5 = 30
6 × 6 = 36
6 × 7 = 42
→ Answer: 30, 36, 42
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EXERCISE 2
(a) 58, 68, 63, 73, 68, ___, ___, ___
Look at changes:
58 → 68 (+10)
68 → 63 (-5)
63 → 73 (+10)
73 → 68 (-5)
Pattern: +10, -5, +10, -5... so next is +10, then -5, then +10
68 + 10 = 78
78 - 5 = 73
73 + 10 = 83
→ Answer: 78, 73, 83
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(b) 7, 8, 15, 23, 38, ___, ___, ___
Look:
7 + 8 = 15
8 + 15 = 23
15 + 23 = 38
This is Fibonacci-style: each number is sum of previous two.
Next:
23 + 38 = 61
38 + 61 = 99
61 + 99 = 160
→ Answer: 61, 99, 160
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(c) 55, 34, 21, 13, 8, ___, ___, ___
Look:
55 - 34 = 21
34 - 21 = 13
21 - 13 = 8
So subtract previous from current to get next? Wait — actually:
It’s going down:
55 → 34 (subtract 21)
34 → 21 (subtract 13)
21 → 13 (subtract 8)
13 → 8 (subtract 5)
Wait — let’s check differences between terms:
Actually, this looks like reverse Fibonacci:
55, 34, 21, 13, 8 — these are Fibonacci numbers backwards!
Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...
So after 8 comes 5, then 3, then 2.
Check:
8 - 5 = 3? No — we’re listing them descending.
Sequence: 55, 34, 21, 13, 8 → next should be 5, then 3, then 2.
Because 8 = 5 + 3, but here we’re just continuing the Fibonacci sequence backward.
Yes: 8 → 5 → 3 → 2
→ Answer: 5, 3, 2
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(d) 576, 288, 144, 72, ___, ___, ___
Look:
576 ÷ 2 = 288
288 ÷ 2 = 144
144 ÷ 2 = 72
So divide by 2 each time:
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
→ Answer: 36, 18, 9
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(e) 47, 40, 45, 38, 43, 36, 41, ___, ___, ___
Look at pairs or alternates:
Odd positions: 47, 45, 43, 41 → decreasing by 2
Even positions: 40, 38, 36 → also decreasing by 2
So next term is position 8 (even): 36 - 2 = 34
Then position 9 (odd): 41 - 2 = 39
Then position 10 (even): 34 - 2 = 32
Wait — let’s list with indices:
1: 47
2: 40
3: 45
4: 38
5: 43
6: 36
7: 41
8: ? → even → 36 - 2 = 34
9: ? → odd → 41 - 2 = 39
10: ? → even → 34 - 2 = 32
But wait — let’s verify the pattern again:
From 47 to 40: -7
40 to 45: +5
45 to 38: -7
38 to 43: +5
43 to 36: -7
36 to 41: +5
So pattern: -7, +5, -7, +5, -7, +5...
So next: 41 - 7 = 34
Then 34 + 5 = 39
Then 39 - 7 = 32
Same result.
→ Answer: 34, 39, 32
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EXERCISE 3
(a) 2, 3, 5, 7, 11, ___, ___
These are prime numbers!
Primes: 2, 3, 5, 7, 11, next is 13, then 17
→ Answer: 13, 17
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(b) 21, 22, 24, 25, 26, ___, ___
Look:
21 → 22 (+1)
22 → 24 (+2)
24 → 25 (+1)
25 → 26 (+1) — wait, that breaks pattern?
Wait — maybe skip some numbers?
List: 21, 22, 24, 25, 26
Differences: +1, +2, +1, +1 — not clear.
Alternative: Maybe it’s all numbers except multiples of 3?
21 is multiple of 3 — included? Hmm.
Wait — perhaps it’s increasing by 1, then 2, then 1, then 1 — doesn’t make sense.
Another idea: Maybe it’s consecutive integers skipping nothing? But 23 is missing.
21, 22, [skip 23], 24, 25, 26 — why skip 23?
Wait — 23 is prime, others aren’t? 21 no, 22 no, 24 no, 25 no, 26 no — 23 is prime, so maybe they skipped primes? But 2 and 3 were in part (a).
Wait — look again: 21, 22, 24, 25, 26 — what’s missing? 23.
Perhaps it’s numbers that are NOT prime? But 21=3×7, 22=2×11, 24=many, 25=5², 26=2×13 — all composite. 23 is prime — so maybe they’re listing composite numbers starting from 21.
Composite numbers after 20: 21, 22, 24, 25, 26, 27, 28...
So next: 27, 28
→ Answer: 27, 28
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(c) 1, 4, 9, 16, 25, ___, ___
These are perfect squares:
1²=1, 2²=4, 3²=9, 4²=16, 5²=25
Next: 6²=36, 7²=49
→ Answer: 36, 49
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(d) 47, 49, 51, 53, 55, 57, ___, ___
All odd numbers increasing by 2:
47, 49, 51, 53, 55, 57 → next 59, 61
→ Answer: 59, 61
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(e) 1, 8, 27, 64, ___, ___
Cubes:
1³=1, 2³=8, 3³=27, 4³=64
Next: 5³=125, 6³=216
→ Answer: 125, 216
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(f) 53, 59, 61, 67, ___, ___
Prime numbers!
After 53: 59 (next prime), then 61, then 67 — yes, all primes.
Next primes after 67: 71, 73
→ Answer: 71, 73
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Final Answer:
EXERCISE 1
(a) 32, 64, 128
(b) 13, 16, 19
(c) 35, 42, 49
(d) 20000, 200000, 2000000
(e) 30, 36, 42
EXERCISE 2
(a) 78, 73, 83
(b) 61, 99, 160
(c) 5, 3, 2
(d) 36, 18, 9
(e) 34, 39, 32
EXERCISE 3
(a) 13, 17
(b) 27, 28
(c) 36, 49
(d) 59, 61
(e) 125, 216
(f) 71, 73
Parent Tip: Review the logic above to help your child master the concept of sequences worksheet grade.
