Why does this find pythagorean triples work: pick an odd number and two numbers that are subsequent and add to the first's square?

Question

Let $a$ = $2k+1$, $k \in \mathbb{N}$

Let $b + (b+1) = a$

Why is $a^2 + b^2 = (b+1)^2$?

Answer 1

I think you meant $b+(b+1)=a^2$.

Then $$a^2+b^2 = b+(b+1) + b^2 = b^2 + 2b + 1 = (b+1)^2$$