Extending number sequences (1) - Number and Place Value by ... - Free Printable

Let's solve each part of this worksheet step by step.

---

1. Fill in the missing numbers in these sequences:

#### 1. 14, 20, 26, 32, ____, ____
- Pattern: Add 6 each time.
- 14 + 6 = 20
- 20 + 6 = 26
- 26 + 6 = 32
- 32 + 6 = 38
- 38 + 6 = 44
- ✔ Answer: 38, 44

---

#### 2. 47, 54, 61, 68, ____, ____
- Pattern: Add 7 each time.
- 47 + 7 = 54
- 54 + 7 = 61
- 61 + 7 = 68
- 68 + 7 = 75
- 75 + 7 = 82
- ✔ Answer: 75, 82

---

#### 3. 9, 17, ____, 33, ____, 49
- Let’s look at the pattern:
- 9 to 17 = +8
- 17 to ? → let’s assume +8 again → 17 + 8 = 25
- 25 to 33 = +8 → yes!
- 33 to ? = +8 → 33 + 8 = 41
- 41 to 49 = +8 → yes!
- ✔ Answer: 25, 41

---

#### 4. 27, 34, ____, ____, ____
- Pattern: Add 7
- 27 + 7 = 34
- 34 + 7 = 41
- 41 + 7 = 48
- 48 + 7 = 55
- ✔ Answer: 41, 48, 55

---

#### 5. 67, 58, 49, ____, 31, ____
- Pattern: Subtract 9 each time
- 67 - 9 = 58
- 58 - 9 = 49
- 49 - 9 = 40
- 40 - 9 = 31 → matches
- 31 - 9 = 22
- ✔ Answer: 40, 22

---

#### 6. 99, 92, ____, 78, ____, 64
- Pattern: Subtract 7
- 99 - 7 = 92
- 92 - 7 = 85
- 85 - 7 = 78 → matches
- 78 - 7 = 71
- 71 - 7 = 64 → matches
- ✔ Answer: 85, 71

---

#### 7. ____, 42, 36, 30, ____, 18
- Pattern: Subtract 6
- 42 - 6 = 36
- 36 - 6 = 30
- 30 - 6 = 24
- 24 - 6 = 18 → matches
- Now go backward: 42 + 6 = 48
- So sequence is: 48, 42, 36, 30, 24, 18
- ✔ Answer: 48, 24

---

#### 8. ____, ____, 78, 70, 62, 54
- Pattern: Subtract 8
- 78 - 8 = 70
- 70 - 8 = 62
- 62 - 8 = 54 → matches
- So before 78: 78 + 8 = 86
- Before that: 86 + 8 = 94
- Sequence: 94, 86, 78, 70, 62, 54
- ✔ Answer: 94, 86

---

9. Take a 9×9 square. Starting at 1, count on in sevens. Circle or colour the numbers you land on.

We are counting in 7s starting from 1:

- 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85

Now locate these numbers in the grid:

```
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
```

Numbers to circle:
- 1 (top-left)
- 8 (row 1, col 8)
- 15 (row 2, col 6)
- 22 (row 3, col 4)
- 29 (row 4, col 2)
- 36 (row 4, col 9)
- 43 (row 5, col 7)
- 50 (row 6, col 5)
- 57 (row 7, col 3)
- 64 (row 8, col 1)
- 71 (row 8, col 8)
- 78 (row 9, col 6)
- 85 → not in grid (max is 81)

So stop at 78.

✔ Circle: 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78

---

10. What do you notice?

Looking at the circled numbers:
They form a diagonal pattern going down and to the right, but with a slight zigzag due to wrapping around columns.

But more precisely, observe their positions:

- They all lie on a diagonal path: moving down one row and right 7 columns each time.
- But since the grid is 9 columns wide, adding 7 means you move left by 2 columns in the next row (because 7 mod 9 = 7, so 7 steps forward = 7 columns right, which is equivalent to -2 columns in modulo 9).

Wait — actually, in terms of column positions:

| Number | Row | Col |
|--------|-----|-----|
| 1 | 1 | 1 |
| 8 | 1 | 8 |
| 15 | 2 | 6 |
| 22 | 3 | 4 |
| 29 | 4 | 2 |
| 36 | 4 | 9 |
| 43 | 5 | 7 |
| 50 | 6 | 5 |
| 57 | 7 | 3 |
| 64 | 8 | 1 |
| 71 | 8 | 8 |
| 78 | 9 | 6 |

You can see that the column numbers decrease by 2 each time, then wrap around.

This creates a diagonal line that moves down-left across the grid.

✔ What do you notice?
The numbers form a diagonal pattern that goes down and to the left every time you add 7.

---

If you went on, would 100 be in your sequence? How do you know?

Sequence: 1, 8, 15, 22, ..., 78, 85, 92, 99, 106...

General term:
This is an arithmetic sequence:
First term = 1, common difference = 7
So: $ a_n = 1 + (n-1) \times 7 $

Check if 100 is in the sequence:
Is there an integer $ n $ such that $ 1 + 7(n-1) = 100 $?
→ $ 7(n-1) = 99 $
→ $ n-1 = 99 / 7 ≈ 14.142... $ → not an integer

So no, 100 is not in the sequence.

Alternatively:
All numbers in the sequence are of the form: $ 7k + 1 $ for some integer $ k $.
Check: $ 100 \mod 7 = ? $

$ 7 \times 14 = 98 $ → $ 100 - 98 = 2 $ → remainder 2
So $ 100 \equiv 2 \mod 7 $, but we need $ \equiv 1 \mod 7 $

✔ So 100 is not in the sequence because it is not congruent to 1 modulo 7.

---

✔ Final Answers:

1. 38, 44
2. 75, 82
3. 25, 41
4. 41, 48, 55
5. 40, 22
6. 85, 71
7. 48, 24
8. 94, 86

9. Circle: 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78

10. What do you notice?
The numbers form a diagonal pattern moving down and to the left across the grid.
Would 100 be in the sequence?
No, because the sequence consists of numbers that are 1 more than a multiple of 7 (i.e., $ \equiv 1 \mod 7 $), and $ 100 \div 7 = 14 $ remainder 2, so $ 100 \equiv 2 \mod 7 $, not 1.

---

Let me know if you'd like a visual of the circled numbers!