What is the rule for the total number of blocks in building n?

• Building 1: 1 block (Level 1: 1)
• Building 2: $2 + 1 = 3$ blocks (Level 1: 2, Level 2: 1)
• Building 3: $3 + 2 + 1 = 6$ blocks (Level 1: 3, Level 2: 2, Level 3: 1)
• Building 4: $4 + 3 + 2 + 1 = 10$ blocks (Level 1: 4, Level 2: 3, Level 3: 2, Level 4: 1)

The sequence of total blocks is 1, 3, 6, 10. This is the sequence of triangular numbers.

The rule for the total number of blocks in Building $n$ is the sum of integers from 1 to $n$. This can be expressed by the formula:

Where $n$ is the building number.

For example:

• For Building 1 ($n=1$): $\frac{1(1+1)}{2} = \frac{1 \times 2}{2} = 1$
• For Building 2 ($n=2$): $\frac{2(2+1)}{2} = \frac{2 \times 3}{2} = 3$
• For Building 3 ($n=3$): $\frac{3(3+1)}{2} = \frac{3 \times 4}{2} = 6$
• For Building 4 ($n=4$): $\frac{4(4+1)}{2} = \frac{4 \times 5}{2} = 10$

The rule for the total number of blocks in Building $n$ is $\boxed{\frac{n(n+1)}{2}}$.