A school is designing a decorative courtyard pattern using identical square tiles. The design grows step by step.
In Step 1, 4 square tiles are placed to form a simple pattern. In each new step, \(4\) more square tiles are added to the design in a symmetric manner. The pattern continues in the same way for every step.
The figure below shows the arrangement up to a few steps (not drawn to scale).
1. How many squares will be there in step 17, step 45, step 63?
Step 17:
Step 45:
Step 63:
2. Write a general formula:
3. How would the formula change if we want to count the number of vertices of all the squares?