The total number of blocks in Building n (which has n levels) is the sum of integers from 1 to n. Write down the rule for the pattern of numbers.

here are the answers for all the questions:

1. Draw the next building in the sequence.
The next building (Building 4) will have 4 blocks in the bottom layer, 3 blocks in the next layer, 2 blocks in the layer above that, and 1 block at the very top.

$$\begin{array}{c} \square \\ \square \square \\ \square \square \square \\ \square \square \square \square \end{array}$$

2. Count the blocks in level 1 of each building. In the table below, fill in the number of blocks in level 1 for buildings 1, 2, 3 and 4.

5. Use the rule to fill in the number of blocks in level 1 for buildings with 6 levels, 10 levels, 12 levels and 20 levels. Enter this in the table.

The number of blocks in Level 1 for Building $n$ (which has $n$ levels) is $n$.
The completed table for questions 2 and 5 is:

$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Building number} & 1 & 2 & 3 & 4 & 6 & 10 & 12 \\ \hline \text{Blocks in level 1} & \boxed{\textbf{1}} & \boxed{\textbf{2}} & \boxed{\textbf{3}} & \boxed{\textbf{4}} & \boxed{\textbf{6}} & \boxed{\textbf{10}} & \boxed{\textbf{12}} \\ \hline \end{array}$$

For a building with 20 levels, the number of blocks in Level 1 is $\boxed{\textbf{20}}$.

3. Write down the rule for the pattern of numbers.
This refers to the total number of blocks in each building.
The rule for the total number of blocks in Building $n$ is the sum of integers from 1 to $n$.

$$\text{Total blocks} = \boxed{\frac{n(n+1)}{2}}$$

4. Test your rule for buildings 3 and 4 using number sentence.
The rule for total blocks is $\frac{n(n+1)}{2}$.

• For Building 3 ($n=3$): Number sentence: $1+2+3 = 6$ Using the rule: $\frac{3(3+1)}{2} = \frac{3 \times 4}{2} = \frac{12}{2} = \boxed{\textbf{6}}$. The rule holds.

• For Building 4 ($n=4$): Number sentence: $1+2+3+4 = 10$ Using the rule: $\frac{4(4+1)}{2} = \frac{4 \times 5}{2} = \frac{20}{2} = \boxed{\textbf{10}}$. The rule holds.

6. Calculate the number of levels that a building will have if it has 3; 9; 21; 60; 63 blocks in level 1. Enter this in the table.
If a building has $X$ blocks in Level 1, then the building number (and thus the number of levels) is $X$.

• If Level 1 has 3 blocks: Number of levels = $\boxed{\textbf{3}}$.
• If Level 1 has 9 blocks: Number of levels = $\boxed{\textbf{9}}$.
• If Level 1 has 21 blocks: Number of levels = $\boxed{\textbf{21}}$.
• If Level 1 has 60 blocks: Number of levels = $\boxed{\textbf{60}}$.
• If Level 1 has 63 blocks: Number of levels = $\boxed{\textbf{63}}$.