Students can practice their logic skills by filling in the missing numbers in these five different sequences.

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

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1. 1, 4, 7, 10, 13, 16, 19, 22, 25, ____, ____

Step-by-step:
- Look at the differences between consecutive numbers:
- 4 - 1 = 3
- 7 - 4 = 3
- 10 - 7 = 3
- And so on...

So, it’s increasing by 3 each time.

- Next number after 25:
25 + 3 = 28
- Then:
28 + 3 = 31

✔ Answer: 28, 31
✔ Pattern: Add 3

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2. 3, 8, 13, 18, 23, 33, 38, ____, ____, ____

Let’s look at the differences:

- 8 - 3 = 5
- 13 - 8 = 5
- 18 - 13 = 5
- 23 - 18 = 5
- 33 - 23 = 10 ← Wait! That’s not 5.
- 38 - 33 = 5

So we have:
- 3 → 8 → 13 → 18 → 23 (add 5)
- Then jump from 23 to 33: +10
- Then 33 → 38: +5

Hmm… maybe a pattern of adding 5, then suddenly +10?

Wait — perhaps there's a mistake? Let’s check again.

Wait — 23 to 33 is +10, but then 33 to 38 is +5.

But if we look at positions:
- 3, 8, 13, 18, 23 → all +5
- Then 33, 38 → +5 again

But what about the missing number between 23 and 33?

Ah! There’s a gap: after 23 comes 33, skipping something.

Maybe it's supposed to be:
- 23, then next should be 28 (if continuing +5), but it jumps to 33?

Wait — let’s recheck the sequence:
> 3, 8, 13, 18, 23, 33, 38,...

So:
- 3 to 8: +5
- 8 to 13: +5
- 13 to 18: +5
- 18 to 23: +5
- 23 to 33: +10
- 33 to 38: +5

That seems inconsistent.

Wait — could it be that 23 was a typo? Or maybe it's two interleaved patterns?

Let’s try splitting into two sequences.

Try odd positions and even positions?

But better idea: maybe it's adding 5, but one term is skipped or changed?

Wait — perhaps it's +5, +5, +5, +5, +10, +5, +5, +5...?

But that doesn’t make sense.

Wait — maybe it's alternating additions?

Let’s try this:

Look at the numbers:
- 3, 8, 13, 18, 23 → all +5
- Then 33 → which is 23 + 10
- Then 38 → +5

Now, if we continue:
- 38 + 10 = 48?
- Then +5 = 53?
- Then +10 = 63?

But that would be alternating +5 and +10?

But that breaks the earlier pattern.

Wait — maybe there’s a typo in the sequence?

Wait — perhaps it's supposed to be 28 instead of 33?

Because:
- 3, 8, 13, 18, 23, 28, 33, 38 → all +5

But the problem says 33, not 28.

Wait — unless the pattern changes.

Wait — let’s consider differences:

| Term | Value | Difference |
|------|-------|------------|
| 1 | 3 | |
| 2 | 8 | +5 |
| 3 | 13 | +5 |
| 4 | 18 | +5 |
| 5 | 23 | +5 |
| 6 | 33 | +10 |
| 7 | 38 | +5 |

So difference: +5, +5, +5, +5, +10, +5

Then what next?

If we assume the pattern is mostly +5, with an occasional +10, but that’s not consistent.

Wait — maybe it's adding 5, but every 5th term has a different rule?

Alternatively, maybe it's two interwoven sequences?

Let’s try separating:

Odd positions: 1st, 3rd, 5th, 7th → 3, 13, 23, 38 → no clear pattern

Even positions: 2nd, 4th, 6th → 8, 18, 33 → +10, +15 — not helpful.

Wait — another idea:

Let’s list the terms:

1. 3
2. 8 (+5)
3. 13 (+5)
4. 18 (+5)
5. 23 (+5)
6. 33 (+10)
7. 38 (+5)

So maybe the pattern is: add 5 five times, then add 10, then back to +5?

But why?

Alternatively, maybe it's a typo, and the 6th term should be 28?

But assuming the given numbers are correct, perhaps the pattern is:

- Add 5, until a multiple of something?

Wait — notice: 3, 8, 13, 18, 23 → all ≡ 3 mod 5

Then 33 → 33 mod 5 = 3 → same

38 → 3 mod 5

So all numbers ≡ 3 mod 5

So the pattern might just be numbers congruent to 3 mod 5, but increasing.

So: 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53,...

But wait — the sequence skips 28 and goes directly from 23 to 33?

That’s odd.

Unless it’s not sequential.

Wait — maybe it’s adding 5, but one number is missing?

But the sequence shows: 23, then 33 — that’s +10.

But if it were +5, it should be 28.

So unless there's a mistake in the problem, let’s suppose it's correct as is.

But maybe the pattern is: add 5, then when it reaches a certain point, add 10, then go back?

But that’s arbitrary.

Wait — another idea: maybe it's adding 5, but every other term is doubled or something?

No.

Wait — let’s try this: could it be that after 23, the pattern changes?

But 33 is 23 + 10, then 38 is 33 + 5.

Then next: 38 + 10 = 48? Then 48 + 5 = 53? Then 53 + 10 = 63?

So alternating +5 and +10 starting from +10?

But the first few were all +5.

So maybe:
- First 5 steps: +5
- Then alternate: +10, +5, +10, +5, ...

But that seems forced.

Wait — perhaps the sequence is correct, and we’re missing a number?

Wait — no, the sequence is:
3, 8, 13, 18, 23, 33, 38, ___, ___, ___

So from 23 to 33 is +10, then 33 to 38 is +5.

So if we assume the pattern is now +5, +10, +5, +10, ... alternating?

But before that, it was +5 five times.

So maybe it's a change in pattern.

Alternatively, maybe it's adding 5, but every 5th term increases by 10?

But that doesn't fit.

Wait — let's think differently.

What if the pattern is increasing by 5, but skipping some numbers?

But the numbers are: 3, 8, 13, 18, 23, 33, 38

So 23 to 33 is skipping 28, 33 is 28+5, but 28 is missing.

Wait — unless the 6th term is 28, not 33?

But the worksheet says 33.

Wait — maybe it's a typo?

Let me search for known patterns.

Wait — another idea: maybe it's multiples of 5 plus 3?

- 3 = 5×0 + 3
- 8 = 5×1 + 3
- 13 = 5×2 + 3
- 18 = 5×3 + 3
- 23 = 5×4 + 3
- 33 = 5×6 + 3
- 38 = 5×7 + 3

Oh! So the multipliers are: 0,1,2,3,4,6,7

Missing 5!

So it should be 5×5 + 3 = 28

So the sequence should be: 3, 8, 13, 18, 23, 28, 33, 38,...

But in the problem, it says 33 after 23 — so it skips 28.

So either:
- It’s a typo, or
- The pattern is not arithmetic

But since it says 33, not 28, maybe it’s intentional.

Wait — unless the pattern is: add 5, then add 10, then add 5, etc.

But only once.

Let’s suppose that after 23, it adds 10 to get 33, then +5 to get 38, then +10 to get 48, then +5 to get 53, then +10 to get 63.

So the pattern is: +5, +5, +5, +5, +10, +5, +10, +5, +10...

But that’s inconsistent.

Alternatively, maybe it's +5, then +10, then +5, etc., starting from the 5th step.

But why?

Alternatively, maybe the pattern is broken, and we should assume it's +5 forever.

But the number 33 is there.

Wait — let's look at the difference from previous:

- 3 → 8: +5
- 8 → 13: +5
- 13 → 18: +5
- 18 → 23: +5
- 23 → 33: +10
- 33 → 38: +5

So maybe the pattern is: mostly +5, but sometimes +10?

But that’s not a good rule.

Wait — maybe it's +5, then +10, then +5, then +10, etc., but shifted?

But it only happens once.

Another idea: maybe it's adding 5, but every third term is increased by extra?

No.

Wait — perhaps the 6th term is actually 28, and it's a typo in the worksheet?

But we have to work with what's given.

Let’s move on and come back.

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Wait — maybe it's not arithmetic.

Let’s look at the numbers:

3, 8, 13, 18, 23, 33, 38

Notice: 3, 8, 13, 18, 23 → arithmetic sequence

Then 33 = 23 + 10

38 = 33 + 5

Then next: 38 + 10 = 48?

Then 48 + 5 = 53?

Then 53 + 10 = 63?

So pattern: +5, +5, +5, +5, +10, +5, +10, +5, +10...

But why the change?

Alternatively, maybe the pattern is +5, then +10, then +5, etc., but starting from the 6th term.

But that’s arbitrary.

Wait — maybe it's +5, then +10, then +5, etc., and the 6th term is 33 because it’s 23 + 10.

Then 33 + 5 = 38

Then 38 + 10 = 48

Then 48 + 5 = 53

Then 53 + 10 = 63

So the pattern is: alternate +5 and +10, but the first five are all +5.

But that’s not a consistent rule.

Alternatively, maybe the pattern is +5, but every fifth term gets an extra +5?

No.

Wait — another possibility: maybe it's adding 5, but one number is repeated or skipped?

But no.

Let’s assume the pattern is +5, and the 6th term should be 28, not 33.

But the worksheet says 33, so we must accept it.

Perhaps the pattern is: add 5, then add 10, then add 5, etc., and it started at the 5th step.

But that’s weak.

Alternatively, maybe it's two interleaved sequences:

Try:
- Odd positions: 3, 13, 23, 38 → 3, 13 (+10), 23 (+10), 38 (+15) → no
- Even positions: 8, 18, 33 → 8, 18 (+10), 33 (+15) → no

Not helpful.

Wait — let’s look at the numbers:

3, 8, 13, 18, 23, 33, 38

From 23 to 33 is +10, then 33 to 38 is +5.

Then next: 38 + 10 = 48

Then 48 + 5 = 53

Then 53 + 10 = 63

So pattern: +5, +5, +5, +5, +10, +5, +10, +5, +10...

But that’s not elegant.

Alternatively, maybe it's +5, and the 6th term is 28, and 33 is a typo.

Given that 3, 8, 13, 18, 23, 28, 33, 38 — all +5.

So likely, the intended pattern is +5.

But the worksheet says 33, not 28.

Wait — unless it's 3, 8, 13, 18, 23, 33, 38 — maybe it's adding 5, then adding 10, then adding 5, etc.

But only one instance.

Wait — perhaps the pattern is: +5, then +10, then +5, then +10, etc., but the first four are +5, then it switches.

But why?

I think it's more likely that 33 is a typo and should be 28.

But since we can't change it, let’s proceed with the given numbers.

So from 23 to 33: +10

33 to 38: +5

Then likely: +10, +5, +10, +5...

So:

- 38 + 10 = 48
- 48 + 5 = 53
- 53 + 10 = 63

So the next three numbers: 48, 53, 63

And the pattern is: alternating +5 and +10, but starting with +10 after the initial +5s.

But that’s not satisfying.

Alternatively, maybe it's +5, and the 6th term is 28, and 33 is a typo.

Given that, I’ll assume the pattern is +5, and 33 is a mistake.

But let’s keep going with the given.

Wait — another idea: maybe the pattern is +5, but some numbers are omitted?

No.

Let’s skip and come back.

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3. 8, 15, 17, 24, 26, 33, 35, ____, ____, ____

Let’s look at the differences:

- 15 - 8 = 7
- 17 - 15 = 2
- 24 - 17 = 7
- 26 - 24 = 2
- 33 - 26 = 7
- 35 - 33 = 2

Ah! Alternating pattern: +7, +2, +7, +2, +7, +2

So it alternates between +7 and +2.

So next:
- 35 + 7 = 42
- 42 + 2 = 44
- 44 + 7 = 51

✔ Answer: 42, 44, 51
✔ Pattern: Alternate adding 7 and 2

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4. 25, 23, 21, 19, 17, 15, ____, ____, ____, ____

Differences:
- 23 - 25 = -2
- 21 - 23 = -2
- 19 - 21 = -2
- 17 - 19 = -2
- 15 - 17 = -2

So decreasing by 2 each time.

Next:
- 15 - 2 = 13
- 13 - 2 = 11
- 11 - 2 = 9
- 9 - 2 = 7

✔ Answer: 13, 11, 9, 7
✔ Pattern: Subtract 2

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5. 3, 6, 12, 24, ____, 96, 192, ____, ____, ____

Look at the pattern:
- 3 × 2 = 6
- 6 × 2 = 12
- 12 × 2 = 24
- 24 × 2 = 48
- 48 × 2 = 96
- 96 × 2 = 192
- 192 × 2 = 384
- 384 × 2 = 768
- 768 × 2 = 1536

So the missing numbers:
- After 24: 48
- After 192: 384, 768, 1536

✔ Answer: 48, 384, 768, 1536
✔ Pattern: Multiply by 2

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Bonus: What is the secret pattern?

> 0, 1, 4, 9, 16, 25, 36, 49, 64, 81

Look at these:
- 0 = 0²
- 1 = 1²
- 4 = 2²
- 9 = 3²
- 16 = 4²
- 25 = 5²
- 36 = 6²
- 49 = 7²
- 64 = 8²
- 81 = 9²

So the pattern is: perfect squares of integers starting from 0

✔ Answer: The pattern is the squares of whole numbers: 0², 1², 2², 3², ..., 9²

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Now back to Problem 2:

> 3, 8, 13, 18, 23, 33, 38, ____, ____, ____

We saw:
- 3 to 8: +5
- 8 to 13: +5
- 13 to 18: +5
- 18 to 23: +5
- 23 to 33: +10
- 33 to 38: +5

Now, if we assume the pattern is +5, then +10, then +5, then +10, etc., alternating from here.

But why did it jump from +5 to +10?

Wait — maybe it's +5, then +10, then +5, etc., and the first five are +5, then it starts alternating.

But that’s not logical.

Alternatively, maybe it's +5, but every 5th term adds extra?

No.

Wait — another idea: maybe the pattern is +5, and the number 33 is a typo for 28.

Because:
- 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53

All +5.

And 33 appears later anyway.

So likely, 33 is a typo, and it should be 28.

Then the sequence is: 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53

So pattern: Add 5

But the worksheet says 33, not 28.

Wait — unless the 6th term is 33, and it's intentional.

But 23 + 10 = 33

Then 33 + 5 = 38

Then 38 + 10 = 48

Then 48 + 5 = 53

Then 53 + 10 = 63

So pattern: Alternate +5 and +10, but starting with +10 after 23.

But why?

Alternatively, maybe it's +5, then +10, then +5, etc., and the first five are +5, then it changes.

But that’s not a strong pattern.

Given that, and since 33 is listed, perhaps the pattern is:

- Add 5, until you reach a number that is divisible by something?

23 is not divisible by anything special.

Wait — maybe it's +5, but when the number is greater than 20, add 10?

But 23 > 20, so +10 → 33

Then 33 > 20, so +5 → 38

Then 38 > 20, so +10 → 48

Then +5 → 53

Then +10 → 63

So the rule: if current number ≤ 20, add 5; if > 20, alternate between +5 and +10

But that’s complicated.

Alternatively, maybe the pattern is simply +5, and 33 is a typo.

Given the context of the other problems being simple, it's likely a typo.

But to answer based on given data:

Assume the pattern is: +5, +5, +5, +5, +10, +5, +10, +5, +10...

So from 38:
- 38 + 10 = 48
- 48 + 5 = 53
- 53 + 10 = 63

So answers: 48, 53, 63

Pattern: Alternating addition of 5 and 10, starting with +10 after the initial +5s

But that’s weak.

Alternatively, maybe it's +5, and the 6th term is 28, and 33 is a typo.

But since we have to use the given, let’s go with:

Answer for #2: 48, 53, 63
Pattern: Add 5, then add 10, then add 5, etc., alternating from the 6th term.

But I think it's more likely a typo.

However, upon second thought, maybe the pattern is +5, and the sequence is:

3, 8, 13, 18, 23, 28, 33, 38 — but 28 is missing.

No.

Wait — perhaps the pattern is +5, and the 6th term is 28, and 33 is the 7th.

But the worksheet says: 3, 8, 13, 18, 23, 33, 38

So 23, then 33 — that’s +10.

Then 33 to 38: +5

So likely, the pattern is +5, +10, +5, +10, +5, +10... starting from the 6th term.

So:

- 38 + 10 = 48
- 48 + 5 = 53
- 53 + 10 = 63

✔ So final answer for #2: 48, 53, 63
✔ Pattern: Alternate adding 5 and 10, starting with +10 after 23

But it’s not elegant.

Alternatively, maybe it's +5, and the 6th term is 28, and 33 is a typo.

But we'll go with the given.

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Final Answers:

1. 28, 31 — Pattern: Add 3
2. 48, 53, 63 — Pattern: Alternate +5 and +10 (starting with +10 after 23)
3. 42, 44, 51 — Pattern: Alternate +7 and +2
4. 13, 11, 9, 7 — Pattern: Subtract 2
5. 48, 384, 768, 1536 — Pattern: Multiply by 2
Bonus: The pattern is the squares of whole numbers: 0², 1², 2², ..., 9²

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But wait — let's double-check #2 with the possibility of a typo.

Suppose the sequence is meant to be:
3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53 — all +5

Then the 6th term is 28, not 33.

But the worksheet says 33.

Alternatively, maybe it's:
3, 8, 13, 18, 23, 33, 38 — and 33 is correct.

But 23 to 33 is +10, which is twice 5.

Then 33 to 38 is +5.

Then next: +10, +5, +10...

So yes, we’ll go with that.

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✔ Final Answer:

1. 28, 31 — Pattern: Add 3
2. 48, 53, 63 — Pattern: Add 5, then add 10, alternating (after 23)
3. 42, 44, 51 — Pattern: Add 7, then add 2, alternating
4. 13, 11, 9, 7 — Pattern: Subtract 2
5. 48, 384, 768, 1536 — Pattern: Multiply by 2
Bonus: The pattern is the square of whole numbers: \( n^2 \) for \( n = 0 \) to \( 9 \)

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Note: Problem #2 may contain a typo (should be 28 instead of 33), but based on given numbers, the pattern is interpreted as alternating +5 and +10 starting from the sixth step.