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 exercise step by step. We’ll look at the pattern in each sequence and figure out what comes next.
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EXERCISE 1
We need to find the next three numbers in each sequence.
(a) 2, 4, 8, 16, ___, ___, ___
Look:
2 → 4 (×2)
4 → 8 (×2)
8 → 16 (×2)
So it’s multiplying by 2 each time.
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 → 4 (+3)
4 → 7 (+3)
7 → 10 (+3)
Adding 3 each time.
Next:
10 + 3 = 13
13 + 3 = 16
16 + 3 = 19
Answer: 13, 16, 19
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(c) 7, 14, 21, 28, ___, ___, ___
Look:
7 → 14 (+7)
14 → 21 (+7)
21 → 28 (+7)
Adding 7 each time — or multiples of 7!
Next:
28 + 7 = 35
35 + 7 = 42
42 + 7 = 49
Answer: 35, 42, 49
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(d) 2, 20, 200, 2000, ___, ___, ___
Look:
2 → 20 (×10)
20 → 200 (×10)
200 → 2000 (×10)
Multiplying by 10 each time.
Next:
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 → 12 (+6)
12 → 18 (+6)
18 → 24 (+6)
Adding 6 each time — multiples of 6!
Next:
24 + 6 = 30
30 + 6 = 36
36 + 6 = 42
Answer: 30, 36, 42
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EXERCISE 2
Again, find next three numbers.
(a) 58, 68, 63, 73, 68, ___, ___, ___
Look carefully:
58 → 68 (+10)
68 → 63 (-5)
63 → 73 (+10)
73 → 68 (-5)
Pattern: +10, -5, +10, -5...
So next:
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
Each number is sum of previous two!
This is like Fibonacci.
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 each number is difference of previous two? Wait — actually:
It’s going down:
55 → 34 (subtract 21?) Not clear.
Wait — reverse Fibonacci?
Actually:
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:
13 - 8 = 5
8 - 5 = 3
5 - 3 = 2
Yes! So subtracting previous from current? Actually, each term is previous minus the one before that? Let's see:
From 55 to 34: 55 - 21 = 34? No.
Better way: This is decreasing Fibonacci.
After 8, next should be 5 (since 13-8=5), then 3 (8-5=3), then 2 (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
Dividing by 2 each time.
Next:
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:
47 → 40 (-7)
40 → 45 (+5)
45 → 38 (-7)
38 → 43 (+5)
43 → 36 (-7)
36 → 41 (+5)
Pattern: -7, +5, -7, +5, -7, +5...
So next:
41 - 7 = 34
34 + 5 = 39
39 - 7 = 32
Answer: 34, 39, 32
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EXERCISE 3
Write next TWO numbers.
(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, not consistent.
Wait — maybe skip some? Or perhaps it’s increasing but skipping primes? No.
Another idea: Maybe it’s all numbers except multiples of 3? Let’s check:
21 (div by 3) — included? Hmm.
Wait — let’s list positions:
Position 1: 21
2: 22
3: 24
4: 25
5: 26
Differences: +1, +2, +1, +1 — doesn’t help.
Wait — maybe it’s consecutive integers starting from 21, but missing 23? Why?
21, 22, [skip 23], 24, 25, 26 — so why skip 23? 23 is prime.
But 21 isn't prime. Maybe not.
Alternative: Perhaps it’s just counting up, and we misread? But 23 is missing.
Wait — look again: 21, 22, 24, 25, 26 — so after 22 jumps to 24. Then continues.
Maybe the pattern is: add 1, add 2, add 1, add 1, then add 2? Not clear.
Wait — another thought: Maybe it’s numbers that are NOT prime? But 23 is prime, so skipped? But 21 is not prime, 22 not, 24 not, etc.
Actually, 23 is the only prime between 21 and 26. So if they’re listing non-prime numbers?
Non-primes from 21: 21, 22, 24, 25, 26, 27, 28...
Yes! That fits.
So next two: 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, so 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.
57 + 2 = 59
59 + 2 = 61
Answer: 59, 61
---
(e) 1, 8, 27, 64, ___, ___
These are cubes!
1³=1, 2³=8, 3³=27, 4³=64, so next:
5³=125, 6³=216
Answer: 125, 216
---
(f) 53, 59, 61, 67, ___, ___
These are prime numbers!
After 67: next primes are 71, 73
Check: 68,69,70 not prime; 71 is prime, 72 no, 73 yes.
Answer: 71, 73
---
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
---
EXERCISE 1
We need to find the next three numbers in each sequence.
(a) 2, 4, 8, 16, ___, ___, ___
Look:
2 → 4 (×2)
4 → 8 (×2)
8 → 16 (×2)
So it’s multiplying by 2 each time.
Next:
16 × 2 = 32
32 × 2 = 64
64 × 2 = 128
Answer: 32, 64, 128
---
(b) 1, 4, 7, 10, ___, ___, ___
Look:
1 → 4 (+3)
4 → 7 (+3)
7 → 10 (+3)
Adding 3 each time.
Next:
10 + 3 = 13
13 + 3 = 16
16 + 3 = 19
Answer: 13, 16, 19
---
(c) 7, 14, 21, 28, ___, ___, ___
Look:
7 → 14 (+7)
14 → 21 (+7)
21 → 28 (+7)
Adding 7 each time — or multiples of 7!
Next:
28 + 7 = 35
35 + 7 = 42
42 + 7 = 49
Answer: 35, 42, 49
---
(d) 2, 20, 200, 2000, ___, ___, ___
Look:
2 → 20 (×10)
20 → 200 (×10)
200 → 2000 (×10)
Multiplying by 10 each time.
Next:
2000 × 10 = 20,000
20,000 × 10 = 200,000
200,000 × 10 = 2,000,000
Answer: 20000, 200000, 2000000
---
(e) 6, 12, 18, 24, ___, ___, ___
Look:
6 → 12 (+6)
12 → 18 (+6)
18 → 24 (+6)
Adding 6 each time — multiples of 6!
Next:
24 + 6 = 30
30 + 6 = 36
36 + 6 = 42
Answer: 30, 36, 42
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EXERCISE 2
Again, find next three numbers.
(a) 58, 68, 63, 73, 68, ___, ___, ___
Look carefully:
58 → 68 (+10)
68 → 63 (-5)
63 → 73 (+10)
73 → 68 (-5)
Pattern: +10, -5, +10, -5...
So next:
68 + 10 = 78
78 - 5 = 73
73 + 10 = 83
Answer: 78, 73, 83
---
(b) 7, 8, 15, 23, 38, ___, ___, ___
Look:
7 + 8 = 15
8 + 15 = 23
15 + 23 = 38
Each number is sum of previous two!
This is like Fibonacci.
Next:
23 + 38 = 61
38 + 61 = 99
61 + 99 = 160
Answer: 61, 99, 160
---
(c) 55, 34, 21, 13, 8, ___, ___, ___
Look:
55 - 34 = 21
34 - 21 = 13
21 - 13 = 8
So each number is difference of previous two? Wait — actually:
It’s going down:
55 → 34 (subtract 21?) Not clear.
Wait — reverse Fibonacci?
Actually:
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:
13 - 8 = 5
8 - 5 = 3
5 - 3 = 2
Yes! So subtracting previous from current? Actually, each term is previous minus the one before that? Let's see:
From 55 to 34: 55 - 21 = 34? No.
Better way: This is decreasing Fibonacci.
After 8, next should be 5 (since 13-8=5), then 3 (8-5=3), then 2 (5-3=2).
Answer: 5, 3, 2
---
(d) 576, 288, 144, 72, ___, ___, ___
Look:
576 ÷ 2 = 288
288 ÷ 2 = 144
144 ÷ 2 = 72
Dividing by 2 each time.
Next:
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
Answer: 36, 18, 9
---
(e) 47, 40, 45, 38, 43, 36, 41, ___, ___, ___
Look at pairs:
47 → 40 (-7)
40 → 45 (+5)
45 → 38 (-7)
38 → 43 (+5)
43 → 36 (-7)
36 → 41 (+5)
Pattern: -7, +5, -7, +5, -7, +5...
So next:
41 - 7 = 34
34 + 5 = 39
39 - 7 = 32
Answer: 34, 39, 32
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EXERCISE 3
Write next TWO numbers.
(a) 2, 3, 5, 7, 11, ___, ___
These are prime numbers!
Primes: 2, 3, 5, 7, 11, next is 13, then 17.
Answer: 13, 17
---
(b) 21, 22, 24, 25, 26, ___, ___
Look:
21 → 22 (+1)
22 → 24 (+2)
24 → 25 (+1)
25 → 26 (+1) — wait, not consistent.
Wait — maybe skip some? Or perhaps it’s increasing but skipping primes? No.
Another idea: Maybe it’s all numbers except multiples of 3? Let’s check:
21 (div by 3) — included? Hmm.
Wait — let’s list positions:
Position 1: 21
2: 22
3: 24
4: 25
5: 26
Differences: +1, +2, +1, +1 — doesn’t help.
Wait — maybe it’s consecutive integers starting from 21, but missing 23? Why?
21, 22, [skip 23], 24, 25, 26 — so why skip 23? 23 is prime.
But 21 isn't prime. Maybe not.
Alternative: Perhaps it’s just counting up, and we misread? But 23 is missing.
Wait — look again: 21, 22, 24, 25, 26 — so after 22 jumps to 24. Then continues.
Maybe the pattern is: add 1, add 2, add 1, add 1, then add 2? Not clear.
Wait — another thought: Maybe it’s numbers that are NOT prime? But 23 is prime, so skipped? But 21 is not prime, 22 not, 24 not, etc.
Actually, 23 is the only prime between 21 and 26. So if they’re listing non-prime numbers?
Non-primes from 21: 21, 22, 24, 25, 26, 27, 28...
Yes! That fits.
So next two: 27, 28
Answer: 27, 28
---
(c) 1, 4, 9, 16, 25, ___, ___
These are perfect squares!
1²=1, 2²=4, 3²=9, 4²=16, 5²=25, so 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.
57 + 2 = 59
59 + 2 = 61
Answer: 59, 61
---
(e) 1, 8, 27, 64, ___, ___
These are cubes!
1³=1, 2³=8, 3³=27, 4³=64, so next:
5³=125, 6³=216
Answer: 125, 216
---
(f) 53, 59, 61, 67, ___, ___
These are prime numbers!
After 67: next primes are 71, 73
Check: 68,69,70 not prime; 71 is prime, 72 no, 73 yes.
Answer: 71, 73
---
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 sequence patterns worksheet.
