Prove by induction $(x+y)^n ≥ \frac{x^{n+1}-y^{n+1}}{x-y}$

Question

Suppose that $x,y\in\Bbb R$ with $x>y>0$. Prove by induction that $$(x+y)^n ≥ \frac{x^{n+1}-y^{n+1}}{x-y}\ \forall n\in\Bbb N$$

How do I prove starting from $n=1$ as I won't be able to get 1 because I don't know $x$ and $y$ value

Answer 1

If $n=1$ then the RHS is $(x+y)^1$ and the LHS is;

$$\frac{(x^2-y^2)}{(x-y)} = \frac{(x+y)(x-y)}{(x-y)} = x+y$$

Answer 2

Substituting n = 0 then

$$ (x+y)^0 \geq \frac{x^1 - y^1}{x - y} $$

Which gives $ 1 \geq 1 $ which is true.