Complete the patterns worksheet with numerical and alphabetical sequences.

Let's solve each pattern step by step and explain the reasoning behind each.

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(1) 2A, 4C, 6E, 8G, ________, ________

- Numbers: 2, 4, 6, 8 → increasing by 2 → next: 10, 12
- Letters: A, C, E, G → every other letter (A=1, C=3, E=5, G=7) → next: I (9), K (11)

→ So: 10I, 12K

✔ Answer: 10I, 12K

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(2) 1, 2, 4, 7, 11, ________, ________

Look at differences:
- 2 - 1 = 1
- 4 - 2 = 2
- 7 - 4 = 3
- 11 - 7 = 4

So differences increase by 1: +1, +2, +3, +4 → next difference: +5 → 11 + 5 = 16
Then +6 → 16 + 6 = 22

✔ Answer: 16, 22

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(3) 10, 20, 30, 40, ________, ________

Clearly increasing by 10 → next: 50, 60

✔ Answer: 50, 60

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(4) A, D, G, J, ________, ________

Letters: A(1), D(4), G(7), J(10) → increasing by 3 positions in alphabet.

Next: J+3 = M(13), then P(16)

✔ Answer: M, P

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(5) 1, 4, 9, 16, ________, ________

Perfect squares: 1², 2², 3², 4² → next: 5² = 25, 6² = 36

✔ Answer: 25, 36

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(6) 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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(7) E, J, O, T, ________

Letters: E(5), J(10), O(15), T(20) → increasing by 5

Next: 25 → Y

✔ Answer: Y

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(8) 7, 9, 11, 13, ________, ________

Odd numbers increasing by 2 → next: 15, 17

✔ Answer: 15, 17

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(9) 4, 10, 16, 22, ________, ________

Differences: 10-4=6, 16-10=6, 22-16=6 → arithmetic sequence with +6

Next: 22+6=28, 28+6=34

✔ Answer: 28, 34

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(10) 51, 44, 37, 30, ________, ________

Decreasing by 7: 51-7=44, 44-7=37, 37-7=30 → next: 30-7=23, 23-7=16

✔ Answer: 23, 16

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(11) 0, 4, 18, 48, ________, ________

Let’s look at the pattern:

- 0 = 0 × 1
- 4 = 1 × 4
- 18 = 2 × 9
- 48 = 3 × 16

Wait — better to think of it as:

Try: n³ + n²? Let’s test:

n=0: 0³ + 0² = 0 ✔
n=1: 1 + 1 = 2 ✘ → not matching

Try: n²(n+1)?
n=0: 0×1=0 ✔
n=1: 1×2=2 ✘ → no

Wait:

Check values:
0, 4, 18, 48

Try:
0 = 1×0
4 = 2×2
18 = 3×6
48 = 4×12

Now: 0, 2, 6, 12 → differences: +2, +4, +6 → next: +8 → 20, then +10 → 30

So:
5×20 = 100
6×30 = 180

But let’s see if there's a simpler formula.

Try: n³ + n²
n=0: 0 + 0 = 0 ✔
n=1: 1 + 1 = 2 ✘
No

Try: n²(n+1):
n=0: 0 → ✔
n=1: 1×2 = 2 ✘
Still no

Wait:
0 = 0 × 1
4 = 1 × 4
18 = 2 × 9
48 = 3 × 16

Ah! The second factor is 1, 4, 9, 16 → squares: 1², 2², 3², 4²

And first factor: 0, 1, 2, 3 → so pattern: n × (n+1)²?

Wait:
For n=0: 0 × (1)² = 0 ✔
n=1: 1 × (2)² = 1×4 = 4 ✔
n=2: 2 × (3)² = 2×9 = 18 ✔
n=3: 3 × (4)² = 3×16 = 48 ✔
So yes!

So next: n=4 → 4 × (5)² = 4×25 = 100
n=5 → 5 × (6)² = 5×36 = 180

✔ Answer: 100, 180

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(12) AZ, BY, CX, DW, ________, ________

First letters: A, B, C, D → increasing by 1 → next: E, F
Last letters: Z, Y, X, W → decreasing by 1 → next: V, U

So: EV, FU

✔ Answer: EV, FU

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(13) 22, 30, 38, 46, ________, ________

Difference: 30-22=8, 38-30=8, 46-38=8 → +8 each time

Next: 46+8=54, 54+8=62

✔ Answer: 54, 62

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(14) 0, 3, 8, 15, ________, ________

Look at numbers:
0 = 1² - 1
3 = 2² - 1 = 4 - 1 = 3 ✔
8 = 3² - 1 = 9 - 1 = 8 ✔
15 = 4² - 1 = 16 - 1 = 15 ✔

So pattern: n² - 1 for n = 1,2,3,4,...

Next: n=5 → 25 - 1 = 24
n=6 → 36 - 1 = 35

✔ Answer: 24, 35

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(15) 100, 95, 85, 70, ________, ________

Differences:
95 - 100 = -5
85 - 95 = -10
70 - 85 = -15

So decreasing by 5 more each time: -5, -10, -15 → next: -20 → 70 - 20 = 50
Then -25 → 50 - 25 = 25

✔ Answer: 50, 25

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✔ Final Answers:

1. 10I, 12K
2. 16, 22
3. 50, 60
4. M, P
5. 25, 36
6. 125, 216
7. Y
8. 15, 17
9. 28, 34
10. 23, 16
11. 100, 180
12. EV, FU
13. 54, 62
14. 24, 35
15. 50, 25

Let me know if you'd like these explained in another way or visualized!