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

I will solve this problem by analyzing each section of the worksheet to find the formula for the number of matchsticks in the nth pattern. I'll start by examining Section A and then proceed through each section systematically.

Section A: Squares in a Row

First, let's count the matchsticks in each pattern:
- Pattern 1 (n=1): 4 matchsticks (one square)
- Pattern 2 (n=2): 7 matchsticks (two squares sharing one side)
- Pattern 3 (n=3): 10 matchsticks (three squares in a row)
- Pattern 4 (n=4): 13 matchsticks (four squares in a row)

The pattern increases by 3 matchsticks each time. This is an arithmetic sequence with first term 4 and common difference 3.
The formula for the nth term is: 4 + (n-1) × 3 = 3n + 1

Section B: Triangles on Top of Squares

Let's count the matchsticks:
- Pattern 1 (n=1): 7 matchsticks (1 triangle + 1 square, sharing one side)
- Pattern 2 (n=2): 12 matchsticks (2 triangles + 2 squares)
- Pattern 3 (n=3): 17 matchsticks (3 triangles + 3 squares)

The pattern increases by 5 matchsticks each time. This is an arithmetic sequence with first term 7 and common difference 5.
The formula for the nth term is: 7 + (n-1) × 5 = 5n + 2

Section C: Triangles Above and Below Squares

Let's count the matchsticks:
- Pattern 1 (n=1): 8 matchsticks (1 square with 1 triangle above and 1 below)
- Pattern 2 (n=2): 14 matchsticks (2 squares with 2 triangles above and 2 below)
- Pattern 3 (n=3): 20 matchsticks (3 squares with 3 triangles above and 3 below)

The pattern increases by 6 matchsticks each time. This is an arithmetic sequence with first term 8 and common difference 6.
The formula for the nth term is: 8 + (n-1) × 6 = 6n + 2

Section D: Staircase Pattern

Let's count the matchsticks:
- Pattern 1 (n=1): 10 matchsticks
- Pattern 2 (n=2): 16 matchsticks
- Pattern 3 (n=3): 22 matchsticks

The pattern increases by 6 matchsticks each time. This is an arithmetic sequence with first term 10 and common difference 6.
The formula for the nth term is: 10 + (n-1) × 6 = 6n + 4

Section E: Hexagons in a Row

Let's count the matchsticks:
- Pattern 1 (n=1): 6 matchsticks (one hexagon)
- Pattern 2 (n=2): 11 matchsticks (two hexagons sharing one side)
- Pattern 3 (n=3): 16 matchsticks (three hexagons in a row)

The pattern increases by 5 matchsticks each time. This is an arithmetic sequence with first term 6 and common difference 5.
The formula for the nth term is: 6 + (n-1) × 5 = 5n + 1

Section F: House Shapes

Let's count the matchsticks:
- Pattern 1 (n=1): 9 matchsticks (one house shape)
- Pattern 2 (n=2): 16 matchsticks (two house shapes sharing one side)
- Pattern 3 (n=3): 23 matchsticks (three house shapes in a row)

The pattern increases by 7 matchsticks each time. This is an arithmetic sequence with first term 9 and common difference 7.
The formula for the nth term is: 9 + (n-1) × 7 = 7n + 2

Extension: Grids

Let's count the matchsticks:
- Pattern 1 (n=1): 4 matchsticks (2x2 grid, which has 4 small squares)
- Pattern 2 (n=2): 12 matchsticks (3x3 grid, which has 9 small squares)
- Pattern 3 (n=3): 24 matchsticks (4x4 grid, which has 16 small squares)
- Pattern 4 (n=4): 40 matchsticks (5x5 grid, which has 25 small squares)
- Pattern 5 (n=5): 60 matchsticks (6x6 grid, which has 36 small squares)

For an n×n grid of small squares, we have (n+1) horizontal lines and (n+1) vertical lines.
Each horizontal line has n matchsticks, so total horizontal matchsticks = (n+1) × n.
Each vertical line has n matchsticks, so total vertical matchsticks = (n+1) × n.
Total matchsticks = 2 × n × (n+1) = 2n² + 2n.

Actually, let me verify this with the given patterns:
- For n=1: 2×1² + 2×1 = 4 ✓
- For n=2: 2×2² + 2×2 = 8 + 4 = 12 ✓
- For n=3: 2×3² + 2×3 = 18 + 6 = 24 ✓
- For n=4: 2×4² + 2×4 = 32 + 8 = 40 ✓
- For n=5: 2×5² + 2×5 = 50 + 10 = 60 ✓

The formula is correct: 2n² + 2n.

## Final Answer:

Here are the formulas for the number of matchsticks in the nth pattern for each section:

Section A: 3n + 1
Section B: 5n + 2
Section C: 6n + 2
Section D: 6n + 4
Section E: 5n + 1
Section F: 7n + 2
Extension: 2n² + 2n