In the problem shown in the image, the pattern starts with a certain number of squares for each step. We are asked to create an equation that describes the relationship between the step number $n$ and the number of squares $p$.

Observing the pattern:

• Step 1: 1 square
• Step 2: 5 squares
• Step 3: 9 squares

We can see that the number of squares increases by 4 each time, which suggests a linear relationship. To describe this pattern:

• The difference between the squares in consecutive steps is constant (5 - 1 = 4, 9 - 5 = 4).
• The pattern grows by 4 squares per step.

Equation derivation:

This linear relationship can be written as: $p = 4n + 1$ Where:

• $p$ is the number of squares in step $n$.
• The coefficient 4 represents the increase in the number of squares at each step.
• The constant 1 represents the number of squares in the first step.

This equation matches what is written in the image.

Would you like to go over the steps of the equation more in detail?

Here are 5 related questions to expand your understanding:

1. How can you verify if the equation $p = 4n + 1$ holds for any step $n$?
2. What happens to the number of squares if $n$ increases by 2 steps instead of 1?
3. Can this pattern be represented using a geometric sequence instead of an arithmetic one?
4. How would the equation change if the pattern started with 3 squares in Step 1 instead of 1?
5. Can you create a similar equation for a different pattern with a different growth rate?

Tip: Identifying a constant rate of increase (like 4 squares per step) suggests a linear relationship, which helps in writing the equation.