Proof of induction of the area formula of a rectangle when its two sides are natural numbers

Question

This is a exercise that I came across while studying a geometry textbook.

Use the method of induction to prove the area formula of a rectangle whose two sides are natural numbers.

I can derive the simple formula of the area of a rectangle by using a method like this:

And so we can derive that $A = xy$ by using $x^2 + y^2 + 2A = (x + y)^2$.

However, I'm having some trouble using induction to prove this. How should I start this problem? Any tips are welcome, thanks.

Answer 1

I believe what is assumed is that

1. You show that if $x,y=1$, then $A(x,y)=1$. (base)
2. You show that if $A(x,1)=1x=x$, then $A(x+1,1)=1(x+1)$. (induction on $x$)
3. You show that if $A(x,y)=xy$, then $A(x,y+1)=x(y+1)$. (induction on $y$)