Quadratic Sequences Worksheet | Cazoom Maths Worksheets - Free Printable

Let's solve this step-by-step, working through each section of the Quadratic Sequences worksheet.

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🔷 Section A

We are given sequences and need to find:
- 6th term
- 7th term
- $ n^\text{th} $ term (formula)

We’ll use second differences to check if it's quadratic and then derive the formula.

#### 🔹 Row 1: 0, 3, 8, 15, 24

Let’s compute first and second differences:

| Term | 0 | 3 | 8 | 15 | 24 |
|------|---|---|---|----|----|
| 1st diff | 3 | 5 | 7 | 9 |
| 2nd diff | 2 | 2 | 2 |

✔ Constant second difference = 2 → Quadratic.

General form: $ a_n = an^2 + bn + c $

Since second difference = 2 ⇒ $ 2a = 2 $ ⇒ $ a = 1 $

Now use values to solve for $ b $ and $ c $:

For $ n=1 $: $ a_1 = 0 $
$$
a(1)^2 + b(1) + c = 0 \Rightarrow 1 + b + c = 0 \Rightarrow b + c = -1 \quad (1)
$$

For $ n=2 $: $ a_2 = 3 $
$$
4a + 2b + c = 3 \Rightarrow 4 + 2b + c = 3 \Rightarrow 2b + c = -1 \quad (2)
$$

Subtract (1) from (2):
$$
(2b + c) - (b + c) = -1 - (-1) \Rightarrow b = 0
$$
Then from (1): $ 0 + c = -1 $ ⇒ $ c = -1 $

So $ a_n = n^2 - 1 $

Check: $ n=1 $: $ 1-1=0 $ ✔, $ n=2 $: $ 4-1=3 $ ✔

Now:
- 6th term: $ 6^2 - 1 = 36 - 1 = 35 $
- 7th term: $ 49 - 1 = 48 $
- $ n^\text{th} $ term: $ n^2 - 1 $

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#### 🔹 Row 2: 3, 6, 11, 18, 27

First differences:
6 - 3 = 3, 11 - 6 = 5, 18 - 11 = 7, 27 - 18 = 9
⇒ 3, 5, 7, 9 → increasing by 2

Second differences: 2, 2, 2 → constant ⇒ quadratic

$ 2a = 2 $ ⇒ $ a = 1 $

Use $ a_n = n^2 + bn + c $

For $ n=1 $: $ a_1 = 3 $
$$
1 + b + c = 3 \Rightarrow b + c = 2 \quad (1)
$$

For $ n=2 $: $ a_2 = 6 $
$$
4 + 2b + c = 6 \Rightarrow 2b + c = 2 \quad (2)
$$

Subtract (1) from (2): $ b = 0 $, then $ c = 2 $

So $ a_n = n^2 + 2 $

Check: $ n=1 $: $ 1+2=3 $ ✔, $ n=2 $: $ 4+2=6 $ ✔

- 6th term: $ 36 + 2 = 38 $
- 7th term: $ 49 + 2 = 51 $
- $ n^\text{th} $ term: $ n^2 + 2 $

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#### 🔹 Row 3: 4, 7, 12, 19, 28

Differences:
7 - 4 = 3, 12 - 7 = 5, 19 - 12 = 7, 28 - 19 = 9
→ 3, 5, 7, 9 → 2nd diff = 2 ⇒ $ a = 1 $

$ a_n = n^2 + bn + c $

n=1: $ 1 + b + c = 4 $ ⇒ $ b + c = 3 $ (1)
n=2: $ 4 + 2b + c = 7 $ ⇒ $ 2b + c = 3 $ (2)

Subtract: $ b = 0 $, $ c = 3 $

So $ a_n = n^2 + 3 $

- 6th term: $ 36 + 3 = 39 $
- 7th term: $ 49 + 3 = 52 $
- $ n^\text{th} $ term: $ n^2 + 3 $

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#### 🔹 Row 4: 1, -2, -7, -14, -23

Differences:
-2 - 1 = -3, -7 - (-2) = -5, -14 - (-7) = -7, -23 - (-14) = -9
→ -3, -5, -7, -9 → 2nd diff = -2 ⇒ $ 2a = -2 $ ⇒ $ a = -1 $

$ a_n = -n^2 + bn + c $

n=1: $ -1 + b + c = 1 $ ⇒ $ b + c = 2 $ (1)
n=2: $ -4 + 2b + c = -2 $ ⇒ $ 2b + c = 2 $ (2)

Subtract: $ b = 0 $, $ c = 2 $

So $ a_n = -n^2 + 2 $

Check: $ n=1 $: $ -1 + 2 = 1 $ ✔, $ n=2 $: $ -4 + 2 = -2 $ ✔

- 6th term: $ -36 + 2 = -34 $
- 7th term: $ -49 + 2 = -47 $
- $ n^\text{th} $ term: $ -n^2 + 2 $

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#### 🔹 Row 5: Given $ n^\text{th} $ term = $ n^2 - 6 $

So:
- 6th term: $ 36 - 6 = 30 $
- 7th term: $ 49 - 6 = 43 $
- $ n^\text{th} $ term: $ n^2 - 6 $

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✔ Section A Summary Table:

| Sequence | 6th term | 7th term | $ n^\text{th} $ term |
|---------|----------|----------|------------------------|
| 0, 3, 8, 15, 24 | 35 | 48 | $ n^2 - 1 $ |
| 3, 6, 11, 18, 27 | 38 | 51 | $ n^2 + 2 $ |
| 4, 7, 12, 19, 28 | 39 | 52 | $ n^2 + 3 $ |
| 1, -2, -7, -14, -23 | -34 | -47 | $ -n^2 + 2 $ |
| — | 30 | 43 | $ n^2 - 6 $ |

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🔷 Section B

Given sequences; find 5th, 6th terms and $ n^\text{th} $ term.

#### 🔹 Row 1: 1, 6, 13, 22

Differences:
6-1=5, 13-6=7, 22-13=9 → 5,7,9 → 2nd diff = 2 ⇒ $ a=1 $

$ a_n = n^2 + bn + c $

n=1: $ 1 + b + c = 1 $ ⇒ $ b + c = 0 $ (1)
n=2: $ 4 + 2b + c = 6 $ ⇒ $ 2b + c = 2 $ (2)

Subtract: $ b = 2 $, $ c = -2 $

So $ a_n = n^2 + 2n - 2 $

Check: $ n=1 $: $ 1 + 2 - 2 = 1 $ ✔, $ n=2 $: $ 4 + 4 - 2 = 6 $ ✔

- 5th term: $ 25 + 10 - 2 = 33 $
- 6th term: $ 36 + 12 - 2 = 46 $
- $ n^\text{th} $ term: $ n^2 + 2n - 2 $

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#### 🔹 Row 2: 5, 10, 17, 26

Diffs: 5, 7, 9 → 2nd diff = 2 ⇒ $ a=1 $

$ a_n = n^2 + bn + c $

n=1: $ 1 + b + c = 5 $ ⇒ $ b + c = 4 $ (1)
n=2: $ 4 + 2b + c = 10 $ ⇒ $ 2b + c = 6 $ (2)

Subtract: $ b = 2 $, $ c = 2 $

So $ a_n = n^2 + 2n + 2 $

- 5th term: $ 25 + 10 + 2 = 37 $
- 6th term: $ 36 + 12 + 2 = 50 $
- $ n^\text{th} $ term: $ n^2 + 2n + 2 $

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#### 🔹 Row 3: 8, 15, 24, 35

Diffs: 7, 9, 11 → 2nd diff = 2 ⇒ $ a=1 $

$ a_n = n^2 + bn + c $

n=1: $ 1 + b + c = 8 $ ⇒ $ b + c = 7 $ (1)
n=2: $ 4 + 2b + c = 15 $ ⇒ $ 2b + c = 11 $ (2)

Subtract: $ b = 4 $, $ c = 3 $

So $ a_n = n^2 + 4n + 3 $

- 5th term: $ 25 + 20 + 3 = 48 $
- 6th term: $ 36 + 24 + 3 = 63 $
- $ n^\text{th} $ term: $ n^2 + 4n + 3 $

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#### 🔹 Row 4: 11, 6, 3, 2

Diffs: 6-11=-5, 3-6=-3, 2-3=-1 → -5, -3, -1 → 2nd diff = 2 ⇒ $ a=1 $

$ a_n = n^2 + bn + c $

n=1: $ 1 + b + c = 11 $ ⇒ $ b + c = 10 $ (1)
n=2: $ 4 + 2b + c = 6 $ ⇒ $ 2b + c = 2 $ (2)

Subtract: $ b = -8 $, $ c = 18 $

So $ a_n = n^2 - 8n + 18 $

- 5th term: $ 25 - 40 + 18 = 3 $
- 6th term: $ 36 - 48 + 18 = 6 $
- $ n^\text{th} $ term: $ n^2 - 8n + 18 $

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#### 🔹 Row 5: Given $ (n+1)^2 + 5 $

Simplify: $ n^2 + 2n + 1 + 5 = n^2 + 2n + 6 $

- 5th term: $ (5+1)^2 + 5 = 36 + 5 = 41 $
- 6th term: $ (6+1)^2 + 5 = 49 + 5 = 54 $
- $ n^\text{th} $ term: $ (n+1)^2 + 5 $

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✔ Section B Summary Table:

| Sequence | 5th term | 6th term | $ n^\text{th} $ term |
|---------|----------|----------|------------------------|
| 1, 6, 13, 22 | 33 | 46 | $ n^2 + 2n - 2 $ |
| 5, 10, 17, 26 | 37 | 50 | $ n^2 + 2n + 2 $ |
| 8, 15, 24, 35 | 48 | 63 | $ n^2 + 4n + 3 $ |
| 11, 6, 3, 2 | 3 | 6 | $ n^2 - 8n + 18 $ |
| — | 41 | 54 | $ (n+1)^2 + 5 $ |

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🔷 Section C

#### 🔹 Row 1: 2, 6, 12, 20

Diffs: 4, 6, 8 → 2nd diff = 2 ⇒ $ a=1 $

$ a_n = n^2 + bn + c $

n=1: $ 1 + b + c = 2 $ ⇒ $ b + c = 1 $ (1)
n=2: $ 4 + 2b + c = 6 $ ⇒ $ 2b + c = 2 $ (2)

Subtract: $ b = 1 $, $ c = 0 $

So $ a_n = n^2 + n $

- 5th term: $ 25 + 5 = 30 $
- 6th term: $ 36 + 6 = 42 $
- $ n^\text{th} $ term: $ n(n+1) $ or $ n^2 + n $

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#### 🔹 Row 2: 1, 3, 6, 10

These are triangular numbers!

Diffs: 2, 3, 4 → 2nd diff = 1 ⇒ $ a = 0.5 $

$ a_n = \frac{1}{2}n^2 + bn + c $

n=1: $ 0.5 + b + c = 1 $ ⇒ $ b + c = 0.5 $ (1)
n=2: $ 2 + 2b + c = 3 $ ⇒ $ 2b + c = 1 $ (2)

Subtract: $ b = 0.5 $, $ c = 0 $

So $ a_n = \frac{1}{2}n^2 + \frac{1}{2}n = \frac{n(n+1)}{2} $

- 5th term: $ \frac{5 \cdot 6}{2} = 15 $
- 6th term: $ \frac{6 \cdot 7}{2} = 21 $
- $ n^\text{th} $ term: $ \frac{n(n+1)}{2} $

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#### 🔹 Row 3: 3, 12, 25, 42

Diffs: 9, 13, 17 → 2nd diff = 4 ⇒ $ a = 2 $

$ a_n = 2n^2 + bn + c $

n=1: $ 2 + b + c = 3 $ ⇒ $ b + c = 1 $ (1)
n=2: $ 8 + 2b + c = 12 $ ⇒ $ 2b + c = 4 $ (2)

Subtract: $ b = 3 $, $ c = -2 $

So $ a_n = 2n^2 + 3n - 2 $

- 5th term: $ 2(25) + 15 - 2 = 50 + 15 - 2 = 63 $
- 6th term: $ 2(36) + 18 - 2 = 72 + 18 - 2 = 88 $
- $ n^\text{th} $ term: $ 2n^2 + 3n - 2 $

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#### 🔹 Row 4: 2, 10, 24, 44

Diffs: 8, 14, 20 → 2nd diff = 6 ⇒ $ a = 3 $

$ a_n = 3n^2 + bn + c $

n=1: $ 3 + b + c = 2 $ ⇒ $ b + c = -1 $ (1)
n=2: $ 12 + 2b + c = 10 $ ⇒ $ 2b + c = -2 $ (2)

Subtract: $ b = -1 $, $ c = 0 $

So $ a_n = 3n^2 - n $

- 5th term: $ 3(25) - 5 = 75 - 5 = 70 $
- 6th term: $ 3(36) - 6 = 108 - 6 = 102 $
- $ n^\text{th} $ term: $ 3n^2 - n $

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#### 🔹 Row 5: Given $ n - 2n^2 + 5 = -2n^2 + n + 5 $

- 5th term: $ -2(25) + 5 + 5 = -50 + 10 = -40 $
- 6th term: $ -2(36) + 6 + 5 = -72 + 11 = -61 $
- $ n^\text{th} $ term: $ -2n^2 + n + 5 $

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✔ Section C Summary Table:

| Sequence | 5th term | 6th term | $ n^\text{th} $ term |
|---------|----------|----------|------------------------|
| 2, 6, 12, 20 | 30 | 42 | $ n^2 + n $ |
| 1, 3, 6, 10 | 15 | 21 | $ \frac{n(n+1)}{2} $ |
| 3, 12, 25, 42 | 63 | 88 | $ 2n^2 + 3n - 2 $ |
| 2, 10, 24, 44 | 70 | 102 | $ 3n^2 - n $ |
| — | -40 | -61 | $ -2n^2 + n + 5 $ |

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🔷 Section D: Patterns

#### 🟨 Pattern 1: Tiles

Patterns:
- P1: 1 tile
- P2: 5 tiles
- P3: 13 tiles
- P4: 25 tiles

Let’s list:
- n=1: 1
- n=2: 5
- n=3: 13
- n=4: 25

Diffs: 4, 8, 12 → 2nd diff = 4 ⇒ $ a = 2 $

So $ a_n = 2n^2 + bn + c $

n=1: $ 2 + b + c = 1 $ ⇒ $ b + c = -1 $ (1)
n=2: $ 8 + 2b + c = 5 $ ⇒ $ 2b + c = -3 $ (2)

Subtract: $ b = -2 $, $ c = 1 $

So $ a_n = 2n^2 - 2n + 1 $

Check:
- n=1: $ 2 - 2 + 1 = 1 $ ✔
- n=2: $ 8 - 4 + 1 = 5 $ ✔
- n=3: $ 18 - 6 + 1 = 13 $ ✔
- n=4: $ 32 - 8 + 1 = 25 $ ✔

✔ So:
1) 10th pattern: $ 2(100) - 2(10) + 1 = 200 - 20 + 1 = 181 $
2) $ n^\text{th} $ pattern: $ 2n^2 - 2n + 1 $

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#### 🔴 Pattern 2: Matches

Each square uses 4 matches, but shared edges reduce total.

Pattern 1: 1 square → 4 matches
Pattern 2: 4 squares → but shared edges → let's count:

- Each internal edge shared → so we can model as grid.

Pattern 1: 1×1 → 4 matches
Pattern 2: 2×2 → 12 matches? Let's draw:

Actually, in matchstick patterns like this, forming a grid:

- For a $ n \times n $ grid of squares:
- Horizontal lines: $ n+1 $ rows × $ n $ segments = $ n(n+1) $
- Vertical lines: $ n+1 $ columns × $ n $ segments = $ n(n+1) $
- Total matches: $ 2n(n+1) $

Check:
- n=1: $ 2(1)(2) = 4 $ ✔
- n=2: $ 2(2)(3) = 12 $ ✔
- n=3: $ 2(3)(4) = 24 $

So:
3) 100th pattern: $ 2 \times 100 \times 101 = 20200 $
4) $ n^\text{th} $ pattern: $ 2n(n+1) $

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✔ Final Answers – Section D

1) Number of tiles in 10th pattern: 181
2) $ n^\text{th} $ pattern: $ 2n^2 - 2n + 1 $
3) Number of matches in 100th pattern: 20,200
4) $ n^\text{th} $ pattern: $ 2n(n+1) $

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✔ Complete Worksheet Answer Summary

#### Section A
| Sequence | 6th term | 7th term | $ n^\text{th} $ term |
|---------|----------|----------|------------------------|
| 0, 3, 8, 15, 24 | 35 | 48 | $ n^2 - 1 $ |
| 3, 6, 11, 18, 27 | 38 | 51 | $ n^2 + 2 $ |
| 4, 7, 12, 19, 28 | 39 | 52 | $ n^2 + 3 $ |
| 1, -2, -7, -14, -23 | -34 | -47 | $ -n^2 + 2 $ |
| — | 30 | 43 | $ n^2 - 6 $ |

#### Section B
| Sequence | 5th term | 6th term | $ n^\text{th} $ term |
|---------|----------|----------|------------------------|
| 1, 6, 13, 22 | 33 | 46 | $ n^2 + 2n - 2 $ |
| 5, 10, 17, 26 | 37 | 50 | $ n^2 + 2n + 2 $ |
| 8, 15, 24, 35 | 48 | 63 | $ n^2 + 4n + 3 $ |
| 11, 6, 3, 2 | 3 | 6 | $ n^2 - 8n + 18 $ |
| — | 41 | 54 | $ (n+1)^2 + 5 $ |

#### Section C
| Sequence | 5th term | 6th term | $ n^\text{th} $ term |
|---------|----------|----------|------------------------|
| 2, 6, 12, 20 | 30 | 42 | $ n^2 + n $ |
| 1, 3, 6, 10 | 15 | 21 | $ \frac{n(n+1)}{2} $ |
| 3, 12, 25, 42 | 63 | 88 | $ 2n^2 + 3n - 2 $ |
| 2, 10, 24, 44 | 70 | 102 | $ 3n^2 - n $ |
| — | -40 | -61 | $ -2n^2 + n + 5 $ |

#### Section D
1) 181
2) $ 2n^2 - 2n + 1 $
3) 20,200
4) $ 2n(n+1) $

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✔ All problems solved! Let me know if you'd like this formatted into a printable answer sheet.