Math worksheet for finding the nth term in sequences using matchstick patterns.

Let's solve each section of this worksheet step by step. The goal is to find the number of matchsticks in the nth pattern for each sequence.

We’ll analyze the patterns, count the matchsticks in the first few terms, look for a rule (sequence), and then find a formula for the nth term.

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🔷 Section A: Squares in a Row

Pattern: Each new square shares a side with the previous one.

- Pattern 1: 1 square → 4 matchsticks
- Pattern 2: 2 squares → 7 matchsticks
- Pattern 3: 3 squares → 10 matchsticks
- Pattern 4: 4 squares → 13 matchsticks

Sequence: 4, 7, 10, 13,...

This increases by 3 each time.

So it’s an arithmetic sequence:
- First term: 4
- Common difference: 3

Formula:
$$
a_n = 3n + 1
$$

✔ Answer for Section A: $ \boxed{3n + 1} $

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🔷 Section B: House Shapes (Triangle on Top of Square)

Each "house" has:
- A square (4 sticks)
- A triangle on top (3 sticks)

But when they are joined side-by-side, they share sides.

Let’s count:

- Pattern 1: 1 house → 4 (square) + 3 (triangle) = 7 sticks
- Pattern 2: 2 houses → Not 14! They share a vertical stick between them.
- Left house: 7 sticks
- Right house: adds only 5 more (because the shared vertical is already counted)
- Total: 7 + 5 = 12
- Pattern 3: Add another house → add 5 more → 12 + 5 = 17

So sequence: 7, 12, 17,...

Increases by 5 each time.

- First term: 7
- Common difference: 5

Formula:
$$
a_n = 5n + 2
$$

Check:
- n=1: 5(1)+2 = 7 ✔
- n=2: 10+2 = 12 ✔
- n=3: 15+2 = 17 ✔

✔ Answer for Section B: $ \boxed{5n + 2} $

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🔷 Section C: Diamond Shapes (Two Triangles Stacked)

Each shape is like a diamond made of two triangles (like a rhombus).

- Pattern 1: 1 diamond → 6 sticks (two triangles sharing a base)
- Pattern 2: 2 diamonds → connected at a point? But look carefully — they're arranged so that each new diamond shares one edge.

Wait, let’s count:

- Pattern 1: 6 sticks
- Pattern 2: Two diamonds sharing a side → total = 6 + 5 = 11 (since one side is shared)
- Pattern 3: Third diamond added → adds 5 more → 16

Sequence: 6, 11, 16,...

Difference: +5

So:
- First term: 6
- Difference: 5

Formula:
$$
a_n = 5n + 1
$$

Check:
- n=1: 5(1)+1 = 6 ✔
- n=2: 10+1 = 11 ✔
- n=3: 15+1 = 16 ✔

✔ Answer for Section C: $ \boxed{5n + 1} $

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🔷 Section D: L-Shaped Squares

Looks like 2x2 blocks but growing horizontally.

- Pattern 1: 2 squares forming an "L" → Let’s count:
- Vertical: 3 sticks
- Horizontal: 3 sticks
- But shared corner? Wait — better to draw it.

Actually, it looks like:

- Pattern 1: 2 squares in an L-shape → How many sticks?

Each square has 4 sides, but shared side reduces total.

For 2 squares sharing one side:
- 4 + 4 – 1 (shared) = 7 sticks? But wait — the L-shape may not be sharing a full side.

Looking at the diagram:

- It's like a corner: 3 vertical and 3 horizontal lines?
- Actually, from image: it's a 2x2 block missing one square? No — it's just two squares forming an L.

But looking closely: the figure shows three squares in an L-shape?

Wait — no. In Section D, first pattern has 2 squares in L-shape.

Let’s count:

- Bottom left square: 4 sticks
- Top right square: shares one side → adds 3 more
- Total: 4 + 3 = 7

Pattern 2: 4 squares? Looks like 2x2 square block.

Wait — actually:

- Pattern 1: 2 squares in L → 7 sticks
- Pattern 2: 4 squares in 2x2 block → how many sticks?

A 2x2 square grid:
- Horizontal lines: 3 rows × 2 columns = 6
- Vertical lines: 3 columns × 2 rows = 6
- Total: 12 sticks

But let's see the pattern:

From image:
- Pattern 1: 2 squares → 7 sticks
- Pattern 2: 4 squares → 12 sticks
- Pattern 3: 6 squares → ? Probably 17?

Wait — maybe not.

Alternatively, perhaps each new “unit” adds 5 sticks?

Let’s re-analyze:

Look again:

- Pattern 1: L-shape with 2 squares → 7 sticks
- Pattern 2: Larger L-shape with 4 squares → appears to be 2×2 block → 12 sticks
- Pattern 3: Even larger → 6 squares? Maybe 3×2 block?

Wait — actually, these look like increasing L-shapes where each step adds a row/column.

But better: count matchsticks directly.

Alternatively, think of it as:

Each new layer adds a "layer" of squares around.

But simpler: look at the growth.

Alternatively, consider:

- Pattern 1: 2 squares → 7 sticks
- Pattern 2: 4 squares → 12 sticks
- Pattern 3: 6 squares → 17 sticks

So: 7, 12, 17 → increases by 5.

So:
- First term: 7
- Common difference: 5

Formula: $ a_n = 5n + 2 $

Check:
- n=1: 5(1)+2 = 7 ✔
- n=2: 10+2 = 12 ✔
- n=3: 15+2 = 17 ✔

But what is the structure?

It seems like each pattern adds 2 more squares and 5 more sticks.

So yes.

✔ Answer for Section D: $ \boxed{5n + 2} $

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🔷 Section E: Hexagons in a Row

Each hexagon has 6 sides.

When joined side-by-side, they share one side.

- Pattern 1: 1 hexagon → 6 sticks
- Pattern 2: 2 hexagons → 6 + 5 = 11 (shared side)
- Pattern 3: 3 hexagons → 11 + 5 = 16

Sequence: 6, 11, 16,...

Common difference: 5

So:
- First term: 6
- Difference: 5

Formula:
$$
a_n = 5n + 1
$$

Check:
- n=1: 5(1)+1 = 6 ✔
- n=2: 10+1 = 11 ✔
- n=3: 15+1 = 16 ✔

✔ Answer for Section E: $ \boxed{5n + 1} $

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🔷 Section F: Zigzag Shapes (Like a Fence)

Each unit looks like a "V" or zigzag.

Let’s count:

- Pattern 1: One "zigzag" → 5 sticks? Let’s see:
- It's a pentagon-like shape? Or a folded shape.

Actually, it's like a "W" shape made of triangles.

But look: each segment is like a "bent" shape.

Pattern 1: 1 unit → 5 sticks
Pattern 2: 2 units → 9 sticks
Pattern 3: 3 units → 13 sticks

Sequence: 5, 9, 13,...

Increases by 4.

So:
- First term: 5
- Difference: 4

Formula:
$$
a_n = 4n + 1
$$

Check:
- n=1: 4(1)+1 = 5 ✔
- n=2: 8+1 = 9 ✔
- n=3: 12+1 = 13 ✔

✔ Answer for Section F: $ \boxed{4n + 1} $

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🔷 Extension: Grids of Squares

Pattern: 1×1, 2×2, 3×3, 4×4 grids.

We need to find number of matchsticks in an $ n \times n $ grid.

Let’s analyze:

#### For a $ n \times n $ grid of squares:

- Number of horizontal lines: $ n+1 $ rows of horizontal lines, each with $ n $ segments → $ n(n+1) $
- Number of vertical lines: $ n+1 $ columns of vertical lines, each with $ n $ segments → $ n(n+1) $

Total matchsticks:
$$
\text{Total} = 2n(n+1)
$$

Let’s test:

- n=1: 2×1×2 = 4 ✔ (1 square → 4 sticks)
- n=2: 2×2×3 = 12 ✔ (2×2 grid → 12 sticks)
- n=3: 2×3×4 = 24 ✔
- n=4: 2×4×5 = 40 ✔

Yes.

So formula:
$$
a_n = 2n(n+1)
$$

Or expanded: $ 2n^2 + 2n $

✔ Answer for Extension: $ \boxed{2n(n+1)} $ or $ \boxed{2n^2 + 2n} $

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✔ Final Answers:

| Section | Formula |
|--------|--------|
| A | $ 3n + 1 $ |
| B | $ 5n + 2 $ |
| C | $ 5n + 1 $ |
| D | $ 5n + 2 $ |
| E | $ 5n + 1 $ |
| F | $ 4n + 1 $ |
| Extension | $ 2n(n+1) $ |

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📝 Summary Explanation:

To find the nth term from a pattern:
1. Count the number of matchsticks in the first few patterns.
2. Look for a pattern in the numbers.
3. Check if it’s arithmetic (constant difference).
4. Use the formula: $ a_n = dn + c $, where $ d $ is the common difference.
5. Plug in $ n=1 $ to find $ c $.
6. For more complex shapes, break into components (horizontal/vertical lines).

This method works well for linear sequences formed by adding shapes with shared sides.

Let me know if you want diagrams or further explanation!