Part (a): Diagram Explanation

The diagram above represents the third square number ($S_3 = 3^2 = 9$) as a $3 \times 3$ array of dots. The additional red dots form a "gnomon" of width 2 around the top and right side of the original square, illustrating the fifth square number ($S_5 = 5^2 = 25$).

Part (b): Pythagorean Relationship

The diagram visually represents a Pythagorean relationship. It shows that adding a double band (the gnomon) to the third square creates the fifth square. This illustrates the formula: $S_5 = S_3 + (2 \times 3) + 2^2$ This relationship can be rewritten in terms of the Pythagorean theorem as: $5^2 = 3^2 + 4^2$ where the gnomon corresponds to the legs of a right triangle ($4$) and the original square is the base ($3$).

Part (c): True or False

The statement is True. If $n^2$ is even, it must be divisible by 4. This is because $n$ must be even for $n^2$ to be even. If $n = 2k$ for some integer $k$, then: $n^2 = (2k)^2 = 4k^2$ Thus, $n^2$ is divisible by 4.

Part (d): Infinite Family of Pythagorean Triples

We can adapt the argument from class, where the difference between two successive square numbers was a square, to now look at the difference between a square number and the square number two steps before it.

Let's assume: $S_{n+2} - S_n = k^2$ By following this logic, we generate Pythagorean triples by considering differences between square numbers that are two steps apart.

I can also generate a labeled diagram similar to part (a) if needed.

Would you like to see more details or ask any questions about the approach?

Here are some related questions:

1. What is the general formula for generating Pythagorean triples?
2. Can we adapt this method to generate triples where both legs are odd?
3. How can we prove the divisibility rule for $n^2$ in more detail?
4. Can we extend this method to non-square relationships?
5. What happens when we increase the gnomon width for larger squares?

Tip: For even natural numbers $n$, always check for divisibility by 4 when working with square numbers.