Question 177
a. Consider the positive integers $a$, $b$, $c$ so that $a \lt b \lt c$ and $a^2 + b^2 = c^2$. If $a_1 = a^2$, $a_2 = ab$, $a_3 = bc$, $a_4 = c^2$, prove that $a_1^2 + a_2^2 + a_3^2 = a_4^2$ and $a_1 \lt a_2 \lt a_3 \lt a_4$.
b. Show that for any $n \in \mathbb{N}$, $n \ge 3$, there exist the positive integers $a_1, a_2, \dots, a_n$ so that $a_1^2 + a_2^2 + \dots + a_{n-1}^2 = a_n^2$ and $a_1 \lt a_2 \lt \dots \lt a_{n-1} \lt a_n$.
b. Show that for any $n \in \mathbb{N}$, $n \ge 3$, there exist the positive integers $a_1, a_2, \dots, a_n$ so that $a_1^2 + a_2^2 + \dots + a_{n-1}^2 = a_n^2$ and $a_1 \lt a_2 \lt \dots \lt a_{n-1} \lt a_n$.