Finding nth Terms from Patterns Worksheet | Fun and Engaging PDF ... - Free Printable

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 with Section A and work my way through each section.

Section A

First, I need to count the matchsticks in each pattern.
- Pattern 1: 4 matchsticks (one square)
- Pattern 2: 7 matchsticks (two squares sharing a side)
- Pattern 3: 10 matchsticks (three squares in a row)
- Pattern 4: 13 matchsticks (four squares in a row)

The sequence is 4, 7, 10, 13...
This is an arithmetic sequence with a common difference of 3.
The formula for the nth term is: 3n + 1

Section B

Now I'll examine Section B.
- Pattern 1: 6 matchsticks (one triangle on top of a square)
- Pattern 2: 11 matchsticks (two triangles on top of two squares)
- Pattern 3: 16 matchsticks (three triangles on top of three squares)

The sequence is 6, 11, 16...
This is an arithmetic sequence with a common difference of 5.
The formula for the nth term is: 5n + 1

Section C

For Section C:
- Pattern 1: 8 matchsticks (one triangle on top and one on bottom of a square)
- Pattern 2: 14 matchsticks (two sets of triangle-square-triangle)
- Pattern 3: 20 matchsticks (three sets of triangle-square-triangle)

The sequence is 8, 14, 20...
This is an arithmetic sequence with a common difference of 6.
The formula for the nth term is: 6n + 2

Section D

Looking at Section D:
- Pattern 1: 7 matchsticks (L-shape made of 3 squares)
- Pattern 2: 12 matchsticks (L-shape made of 5 squares)
- Pattern 3: 17 matchsticks (L-shape made of 7 squares)

The sequence is 7, 12, 17...
This is an arithmetic sequence with a common difference of 5.
The formula for the nth term is: 5n + 2

Section E

For Section E:
- Pattern 1: 6 matchsticks (one hexagon)
- Pattern 2: 11 matchsticks (two hexagons sharing a side)
- Pattern 3: 16 matchsticks (three hexagons in a row)

The sequence is 6, 11, 16...
This is an arithmetic sequence with a common difference of 5.
The formula for the nth term is: 5n + 1

Section F

For Section F:
- Pattern 1: 7 matchsticks (one house shape)
- Pattern 2: 12 matchsticks (two houses sharing a side)
- Pattern 3: 17 matchsticks (three houses in a row)

The sequence is 7, 12, 17...
This is an arithmetic sequence with a common difference of 5.
The formula for the nth term is: 5n + 2

Extension

For the Extension section:
- Pattern 1: 4 matchsticks (2x2 grid)
- Pattern 2: 9 matchsticks (3x3 grid)
- Pattern 3: 16 matchsticks (4x4 grid)
- Pattern 4: 25 matchsticks (5x5 grid)
- Pattern 5: 36 matchsticks (6x6 grid)

The sequence is 4, 9, 16, 25, 36...
These are perfect squares: 2², 3², 4², 5², 6²...
The formula for the nth term is: (n+1)²

## Final Answer:

Section A: 3n + 1
Section B: 5n + 1
Section C: 6n + 2
Section D: 5n + 2
Section E: 5n + 1
Section F: 5n + 2
Extension: (n+1)²