What is the rule for the pattern of triangles, and how many triangles are in the tenth pattern?

• Pattern 1: 3 dots
• Pattern 2: 6 dots
• Pattern 3: 10 dots
• Pattern 4: 15 dots This is a sequence of triangular numbers. The number of dots in pattern $n$ is given by the formula $\frac{(n+1)(n+2)}{2}$.

For the 5th pattern ($n=5$):
Number of dots $= \frac{(5+1)(5+2)}{2} = \frac{6 \times 7}{2} = \frac{42}{2} = 21$.
The 5th pattern will have 21 dots arranged in a triangle with 6 dots on each side.

• Pattern 1: 1 triangle, 3 squares. Total = 4 blocks.
• Pattern 2: 2 triangles, 6 squares. Total = 8 blocks.
• Pattern 3: 3 triangles, 9 squares. Total = 12 blocks.
• Pattern 4: 4 triangles, 12 squares. Total = 16 blocks. The number of blocks in pattern $n$ is $4n$.

For the 5th pattern ($n=5$):
Number of blocks $= 4 \times 5 = 20$.
The 5th pattern will consist of 5 houses, each with 1 triangle and 3 squares, for a total of 5 triangles and 15 squares.

2. Extend the patterns.

• Pattern 1: A $2 \times 2$ square, total 4 small squares.
• Pattern 2: A $4 \times 4$ square, total 16 small squares. The side length of the square is $2^n$. The total number of small squares is $(2^n)^2 = 2^{2n}$.

For the 3rd pattern ($n=3$):
Side length $= 2^3 = 8$.
Total number of small squares $= 8 \times 8 = 64$.
The 3rd pattern will be an *$8