Investigate and demonstrate the physical structure and then complete the table below. Pattern number 4 and 10.

• Pattern 1 (1 triangle): 3 matchsticks
• Pattern 2 (2 triangles): 5 matchsticks
• Pattern 3 (3 triangles): 7 matchsticks The number of matchsticks increases by 2 for each additional triangle. This is an arithmetic progression.

• The "Repeated addition" shows 1 plus 2 added $n$ times, where $n$ is the pattern number.
• "The rule" simplifies this to $1 + (2 \times n)$.

• For Pattern number 4:
  • Number of matchsticks: $7 + 2 = 9$.
  • Repeated addition: $1+2+2+2+2$.
  • The rule: $1+(2 \times 4)$.
• For Pattern number 10:
  • Number of matchsticks: $1 + (2 \times 10) = 1 + 20 = 21$.
  • Repeated addition: $1+2+2+2+2+2+2+2+2+2+2$.
  • The rule: $1+(2 \times 10)$.

The completed table is:

The general rule for the number of matchsticks ($N$) for $n$ triangles is:

The number of matchsticks needed for a pattern with 20 triangles is $\boxed{\text{41}}$.