Number Patterns worksheet for practicing identifying and completing numerical sequences.

Here are the completed number patterns with the missing numbers filled in. I have identified the rule for each pattern (addition or subtraction) to find the missing values.

1. Pattern: Add 5
* Rule: $60 + 5 = 65$, $65 + 5 = 70$. Going backward: $60 - 5 = 55$, etc.
* Sequence: 45, 50, 55, 60, 65, 70, 75, 80

2. Pattern: Subtract 4
* Rule: $28 - 4 = 24$, $24 - 4 = 20$. Going backward: $28 + 4 = 32$, etc.
* Sequence: 40, 36, 32, 28, 24, 20, 16, 12

3. Pattern: Subtract 2
* Rule: $98 - 2 = 96$. Going forward from 62: $62 - 2 = 60$, etc.
* Sequence: 100, 98, 96, 94, 92, 90, 88, 86, 62, 60, 58, 56, 54, 52

4. Pattern: Mixed (Subtract 10, then Add 5)
* Rule: The pattern alternates between subtracting 10 and adding 5.
* Check: $120 - 10 = 110$; $110 + 5 = 115$; $115 - 10 = 105$. This fits.
* Sequence: 130, 120, 110, 115, 105, 100, 105, 95, 100, 85, 75, 80

5. Pattern: Subtract 100
* Rule: Each number is 100 less than the previous one.
* Sequence: 997, 897, 797, 697, 597, 497, 397, 297, 197, 97

6. Pattern: Subtract 3
* Rule: $72 - 3 = 69$; $69 - 3 = 66$... wait, looking at 59 to 56, that is also minus 3. Let's check the gap between 66 and 59. $66 - 7 = 59$. Let's re-examine.
* Actually, let's look closer. $72 \rightarrow \text{blank} \rightarrow \text{blank} \rightarrow 59 \rightarrow 56$.
* $59 - 3 = 56$. So the step is likely 3.
* $72 - 3 = 69$. $69 - 3 = 66$. $66 - 3 = 63$. But the next given number is 59. $63 - 4 = 59$? No.
* Let's check the position. Box, Box, Box, 59, 56.
* If we go backward from 59: $59+3=62$, $62+3=65$, $65+3=68$. Does $68-3=65$? Yes. Does $72-3=69$?
* Let's look at the image again carefully.
* Item 6: `[ ] [ ] [ ] 72 [ ] [ ] 59 56 [ ] [ ]`
* Ah, 72 is given. Then two blanks, then 59, 56.
* $59 - 3 = 56$. So the rule is subtract 3.
* Backward from 72: The box before 72 would be $72+3=75$. The one before that $78$. The one before that $81$.
* Forward from 72: $72-3=69$. $69-3=66$. Next is 59? $66-7=59$. That breaks the pattern.
* Let me re-read the sequence positions for #6.
* Boxes: 1, 2, 3, Value(72), Box, Box, Value(59), Value(56), Box, Box.
* Wait, is it possible the pattern changes? Or did I miscount boxes?
* Let's look at the spacing. `__ __ __ 72 __ __ 59 56 __ __`
* From 59 to 56 is -3.
* From 72 to 59 is a difference of 13. There are 2 boxes between them. Steps: $72 \rightarrow A \rightarrow B \rightarrow 59$.
* If the step is constant, $72 - x - x - x = 59 \Rightarrow 3x = 13$. Not an integer.
* Let's look really closely at crop 5 and 6.
* Crop 5 shows end of #5 and start of #6.
* Crop 6 shows #6 clearly: `[ ] [ ] [ ] 72 [ ] [ ] 59 56 [ ] [ ]`
* Is it possible the number is not 72? It looks like 72.
* Is it possible the number is not 59? It looks like 59.
* Let's try a different pattern. Maybe it's -3, -3, -3, -3...
* If it's -3: $72, 69, 66, 63, 60, 57, 54...$ This doesn't hit 59 or 56.
* Let's try -2: $72, 70, 68, 66, 64, 62, 60, 58, 56$. Hits 56!
* Let's check if it hits 59? No, 59 is odd. 72 is even. You can't get from an even to an odd with steps of 2.
* Let's look at the number "59" again. Could it be 63? No, looks like 59.
* Could "72" be 71? No.
* Let's look at the previous number in the sequence if we assume the pattern is consistent across the whole line.
* What if the pattern is -3 but there is a typo in my reading of "59"?
* Let's look at #7. `__ __ __ 69 __ __ __ 57 54`.
* $57 - 3 = 54$. Rule is -3.
* Backwards from 69 with -3: $69, 72, 75, 78$.
* Forwards from 69 with -3: $66, 63, 60, 57$. Matches perfectly.
* So #7 is definitely -3.
* Now back to #6. `__ __ __ 72 __ __ 59 56 __ __`
* If #7 is -3, maybe #6 is also -3?
* If #6 is -3: $72, 69, 66, 63, 60, 57, 54$.
* The printed numbers are 59 and 56. This is very strange. $59$ and $56$ differ by 3. $72$ and $59$ differ by 13.
* Is it possible the number is 62 instead of 72? $62, 59, 56$. That works with -3.
* Looking at the image, the first digit of 72 has a horizontal top bar. It looks like a 7.
* However, in many of these worksheets, if there's a sequence `A, B, C, D`, and D is 56, C is 59, then B should be 62, A should be 65.
* The number printed is clearly 72.
* Let's look at the boxes before 72. There are 3 boxes.
* Let's look at the boxes after 72. There are 2 boxes before 59.
* Sequence: $X_1, X_2, X_3, 72, Y_1, Y_2, 59, 56, Z_1, Z_2$.
* Difference between 72 and 59 is 13. Steps involved: $72 \to Y_1 \to Y_2 \to 59$. That is 3 steps.
* $13 / 3$ is not an integer.
* Is it possible the pattern is alternating?
* Let's check #4 again. $120, 110, 115, 105, 100, 105, 95, 100, 85$.
* $120-10=110$. $110+5=115$. $115-10=105$. $105-5=100$? No, $105-5=100$. Then $100+5=105$. Then $105-10=95$. Then $95+5=100$. Then $100-15=85$? No.
* Let's re-evaluate #4.
* $120 \xrightarrow{-10} 110 \xrightarrow{+5} 115 \xrightarrow{-10} 105 \xrightarrow{-5} 100$? No.
* Let's look at the printed numbers in #4: `__ 120 __ __ 105 __ __ __ 85 __ __`
* My previous solution for #4 was: $130, 120, 110, 115, 105, 100, 105, 95, 100, 85$.
* Check: $100 \xrightarrow{-15} 85$? No.
* Let's restart #4 logic.
* Given: $120$ at pos 2. $105$ at pos 5. $85$ at pos 9.
* Pos 2 to Pos 5 is 3 steps. $120 \to 105$ is $-15$. Average step $-5$.
* If step is $-5$: $120, 115, 110, 105$. This fits Pos 2 to 5 perfectly.
* Let's test $-5$ for the rest.
* Pos 5 is 105. Pos 9 is 85. Steps: 4 steps. $105 \to 85$ is $-20$. $-20 / 4 = -5$.
* So the pattern for #4 is simply Subtract 5.
* Why did I think it was mixed before? I misread the boxes.
* Correct #4: 125, 120, 115, 110, 105, 100, 95, 90, 85, 80, 75.

Now back to #6 with fresh eyes.
Given: 72 at Pos 4. 59 at Pos 7. 56 at Pos 8.
Pos 7 to Pos 8: $59 \to 56$ is $-3$.
Pos 4 to Pos 7: 3 steps. $72 \to 59$ is $-13$.
This still doesn't divide evenly by 3.
Is it possible the number is 69 at Pos 7? No, looks like 59.
Is it possible the number is 71 at Pos 4? No.
Is it possible the number is 62 at Pos 4? If Pos 4 is 62, then $62 \to 59 \to 56$ is perfect -3s.
Visually, does "72" look like "62"? The top bar is flat. A 6 usually curves. It really looks like 72.
However, in grade school math, typos happen. Or maybe I am missing a complex pattern.
Let's look at #8.
`281 __ __ __ 141 __ __ 56 __ __`
$281 \to 141$. Diff 140. Positions: 1 to 5. 4 steps. $140/4 = 35$.
Let's test Subtract 35.
$281 - 35 = 246$.
$246 - 35 = 211$.
$211 - 35 = 176$.
$176 - 35 = 141$. (Matches Pos 5).
$141 - 35 = 106$.
$106 - 35 = 71$.
$71 - 35 = 36$.
Pos 8 is 56? My calc says 36.
Let's check the position of 56.
Boxes: `281 [1] [2] [3] 141 [4] [5] 56 [6] [7]`
Pos 1: 281. Pos 5: 141. Pos 8: 56.
Step 1-5 (4 jumps): $281-141=140$. $140/4 = 35$.
Step 5-8 (3 jumps): $141-56=85$. $85/3 = 28.33$.
This implies the pattern changes or my reading of "56" is wrong.
Could "56" be 36? The top loop of a 3 can look closed like a 5 in bad print.
If it's 36: $141 \xrightarrow{-35} 106 \xrightarrow{-35} 71 \xrightarrow{-35} 36$. This works perfectly.
So #8 is likely Subtract 35, and the printed "56" is a typo for "36" or I should treat it as 36 based on the strong signal from 281->141.
OR, is the pattern different?

Let's re-read #6. `__ __ __ 72 __ __ 59 56 __ __`
If the pattern is consistent, and 59->56 is -3, then the whole thing should be -3.
If the whole thing is -3, then Pos 4 should be related to Pos 7 by $3 \times -3 = -9$.
$72 - 9 = 63$. But Pos 7 is 59.
$59 + 9 = 68$. But Pos 4 is 72.
Difference is 4.

Let's look at #9.
`__ __ 101 __ __ __ __ __ 89 87 __`
End: $89 \to 87$ is -2.
Let's test Subtract 2.
Pos 9: 89. Pos 10: 87.
Pos 3: 101.
Distance Pos 3 to Pos 9 is 6 steps.
$101 - 89 = 12$.
$12 / 6 = 2$.
So #9 is definitely Subtract 2.
Sequence: 105, 103, 101, 99, 97, 95, 93, 91, 89, 87, 85.

Let's look at #10.
`__ __ __ __ __ __ 1248 1176 78 __ __`
Wait, the numbers are huge then tiny?
`1248`, `1176`, `78`.
$1248 - 1176 = 72$.
$1176 - 78 = 1098$.
This makes no sense.
Let me look at the image for #10 again.
Crop 8: `__ __ __ __ __ __ 1248 1176 78 __ __`
Is it `124`, `117`, `67`? No, commas or spacing?
Ah, look at the spacing in #10.
`__ __ __ __ __ __ 124 8 __ 117 6 __ 78 __ __` ? No.
Let's look at the numbers `1248` and `1176`.
Maybe it's `124`, `8`? No.
Maybe the pattern is Subtract 72?
$1248 - 72 = 1176$.
$1176 - 72 = 1104$.
But the next number is `78`.
Is it possible the number is 1104 and it's printed as `78`? Unlikely.
Is it possible the number is 108? $1176 - 108$? No.

Let's reconsider the digits.
Could `1248` be `12`, `4`, `8`? No, they are grouped.
Could `1176` be `11`, `7`, `6`?

Let's look at the gap between 1176 and 78.
Maybe the number is 1104 and the student/book made a massive typo?
Or maybe the pattern is Divide?
$1248 / 2 = 624$. No.

Let's look really closely at #10 in the original image.
It says: `... __ __ 1248 1176 78 __ __`
Wait, is that a 1 before the 78? `178`?
If it's 1176 -> 1104 -> 1032...
If it's 1176 -> 1098?

Let's try a different pattern for #10.
What if the numbers are `124`, `8`? No.
What if the number is 1104 and it looks like 78 because of bad printing?
Actually, $1176 - 72 = 1104$.
$1104 - 72 = 1032$.

Let's look at the number `78` again.
Is it possible it is 1032? No.
Is it possible the previous number is not 1176?
$1248 - 1176 = 72$.

Let's assume there is a typo in the question for #6, #8, and #10, which is common in these online worksheets. I will provide the most logical mathematical answer assuming a constant arithmetic progression, noting the likely intended values if they deviate.

Re-evaluating #6:
Pattern: Subtract 3.
Intended sequence around the glitch: ..., 65, 62, 59, 56, ...
Printed "72" is likely a typo for "62".
If I must fill the boxes based on the printed "72":
If I force -3: 72, 69, 66, 63 (doesn't match 59).
If I force fit 72 and 59: Step is -4.33. Impossible.
Most likely answer key expects: 81, 78, 75, 72, 69, 66, 59 (Error in problem), 56...
Actually, let's look at the boxes for #6 again.
`[ ] [ ] [ ] 72 [ ] [ ] 59 56 [ ] [ ]`
If the pattern is -3, and we ignore the "72" vs "59" conflict and just follow the local clues:
Left of 72 (subtract 3 backwards = add 3): 75, 78, 81.
Right of 72 (subtract 3): 69, 66.
Then we hit 59. The jump $66 \to 59$ is -7.
Then $59 \to 56$ is -3.
Then $56 \to 53 \to 50$.
This is inconsistent.
However, usually, the "given" numbers define the pattern.
Given: 59, 56. Pattern: -3.
Apply -3 everywhere.
Fill: 81, 78, 75, 72 (Typo for 62?), 69, 66, 59, 56, 53, 50.
I will provide this sequence, assuming the standard pattern holds despite the likely typo in the printed '72'.

Re-evaluating #8:
Pattern: Subtract 35.
Sequence: 281, 246, 211, 176, 141, 106, 71, 36 (Printed as 56?), 1, -34.
I will fill based on Subtract 35.
Fill: 246, 211, 176, 141, 106, 71, 56 (Typo for 36?), 21, -14.
Wait, negative numbers? Grade 3/4 usually doesn't do negatives.
Let's check if the pattern is Subtract 31.5? No.
Let's check if the pattern is Subtract 30?
$281 - 30 = 251$. $251-30=221$. $221-30=191$. $191-30=161 \neq 141$.
Let's check if the pattern is Subtract 35 but the start is different?
What if the number is 281 and 141 is correct, but 56 is correct?
Maybe two different patterns?
Part 1: $281 \to 141$ (-140 over 4 steps = -35).
Part 2: $141 \to 56$ (-85 over 3 steps = -28.3).
This worksheet seems to have significant errors in problems 6, 8, and 10.

Let's try one more interpretation for #10.
`1248`, `1176`, `78`.
Could it be `124`, `8`? No.
Could it be `12`, `48`? No.
What if the number is 1104 and it is misprinted as 78?
Or what if the number is 108?
$1176 - 108 = 1068$.

Let's look at the visual of #10 again.
`1248` `1176` `78`
Is it possible the pattern is Subtract 72?
$1248 - 72 = 1176$.
$1176 - 72 = 1104$.
$1104 - 72 = 1032$.
If the third number is a typo for 1104, then the rest follows.
But it says `78`.
Is it possible the pattern is Divide by something?
$1248 / 1.06...$

Okay, I will solve the ones that are clear (1, 2, 3, 4, 5, 7, 9) with high confidence.
For 6, 8, 10, I will provide the most likely intended arithmetic sequence, noting that the printed numbers contain typos.

Final Checks for Clear Problems:
1. +5: 45, 50, 55, 60, 65, 70, 75, 80. (Correct)
2. -4: 40, 36, 32, 28, 24, 20, 16, 12. (Correct)
3. -2: 100, 98, 96, 94, 92, 90, 88, 86, 62, 60, 58, 56, 54, 52. (Correct)
*Wait*, in #3, there is a jump from 86 to 62?
Image: `__ 98 96 __ __ __ __ __ 62 __ __ __ __ __`
Let's count boxes between 96 and 62.
`96 [1] [2] [3] [4] [5] 62`
5 boxes. 6 steps.
$96 - 62 = 34$. $34 / 6$ is not integer.
Let's re-read #3.
`[ ] 98 96 [ ] [ ] [ ] [ ] [ ] 62 [ ] [ ] [ ] [ ] [ ]`
Steps:
Pos 2: 98. Pos 3: 96. Diff -2.
Pos 9: 62.
Pos 3 to Pos 9 is 6 steps.
$96 - 62 = 34$.
This implies the pattern is NOT constant -2, or there is a typo.
However, $98, 96$ strongly suggests -2.
If it were -2, Pos 9 should be $96 - (6 \times 2) = 96 - 12 = 84$.
The printed number is 62.
Is it possible the number is 84 and looks like 62? No.
Is it possible the number is 96 and the next is 62?
Maybe the pattern is -3?
$98 - 3 = 95 \neq 96$.

Let's look at the other side of #3.
After 62: `[ ] [ ] [ ] [ ] [ ]`
If pattern is -2: 60, 58, 56, 54, 52.

I will stick with -2 for #3 as it's the only logical start, assuming 62 is a typo for 84 or similar. OR, maybe the blank count is different.
Let's recount #3 boxes.
`[ ] 98 96 [ ] [ ] [ ] [ ] [ ] 62`
Maybe the first box is 100.
$100, 98, 96, 94, 92, 90, 88, 86, 84$.
The printed number is 62.
This worksheet is full of typos.

Strategy for Final Answer:
I will provide the answers based on the strongest local pattern (usually the first two or last two numbers) and assume arithmetic progression. I will note discrepancies where obvious.

Refined Answers:
1. 45, 50, 55, 60, 65, 70, 75, 80 (Rule: +5)
2. 40, 36, 32, 28, 24, 20, 16, 12 (Rule: -4)
3. 100, 98, 96, 94, 92, 90, 88, 86, 62 (Likely Typo for 84), 60, 58, 56, 54, 52 (Rule: -2)
4. 125, 120, 115, 110, 105, 100, 95, 90, 85, 80, 75 (Rule: -5)
5. 997, 897, 797, 697, 597, 497, 397, 297, 197, 97 (Rule: -100)
6. 81, 78, 75, 72 (Likely Typo for 62), 69, 66, 59, 56, 53, 50 (Rule: -3)
7. 78, 75, 72, 69, 66, 63, 60, 57, 54, 51, 48 (Rule: -3)
8. 246, 211, 176, 141, 106, 71, 56 (Likely Typo for 36), 21, -14 (Rule: -35) -> *Actually, for a student, I'll stop at positive numbers or adjust.* If 56 is correct, maybe the pattern changes? No, keep it simple. I'll provide the -35 sequence.
9. 105, 103, 101, 99, 97, 95, 93, 91, 89, 87, 85 (Rule: -2)
10. 1464, 1392, 1320, 1248, 1176, 1104, 78 (Likely Typo for 1104 or 1032?), ...
Let's look at #10 again. `1248, 1176`. Diff 72.
Prev: $1176+72=1248$.
Prev: $1248+72=1320$.
Prev: $1320+72=1392$.
Prev: $1392+72=1464$.
Prev: $1464+72=1536$.
Next: $1176-72=1104$.
Next: $1104-72=1032$.
The printed `78` is completely off. I will provide the sequence based on the clear start.

Final Answer:
1. 45, 50, 55, 70, 75, 80
2. 40, 36, 32, 20, 16, 12
3. 100, 94, 92, 90, 88, 86, 60, 58, 56, 54, 52 *(Note: Printed '62' appears to be a typo for '84')*
4. 125, 115, 110, 100, 95, 90, 80, 75
5. 997, 897, 797, 697, 597, 497, 397, 297, 197, 97
6. 81, 78, 75, 69, 66, 53, 50 *(Note: Printed '72' appears to be a typo for '62')*
7. 78, 75, 72, 66, 63, 60, 51, 48
8. 246, 211, 176, 106, 71, 21, -14 *(Note: Printed '56' appears to be a typo for '36')*
9. 105, 103, 99, 97, 95, 93, 91, 85
10. 1536, 1464, 1392, 1320, 1104, 1032 *(Note: Printed '78' appears to be a typo)*