Olympiad problem algebra inequality

Question

I'm having trouble solving the following inequality problem:

If $n$ is positive integer greater than $1$, and $x>y>1$, then show that:

$\frac{x^{n+1}-1}{x(x^{n-1}-1)} > \frac{y^{n+1}-1}{y(y^{n-1}-1)}$

Any hints? Thanks.

Answer 1

Since $x^{n-1} > x^{n-2} > \dots > 1$, and $y^{n-1} > y^{n-2} > \dots > 1$, we have from the rearrangement inequality $$ x^{n-1} y^{n-1} + x^{n-2} y^{n-2} + \dots + 1 > x^{n-1} + x^{n-2} y + \dots + y^{n-1}. $$ Multiplying both side by $(xy - 1)(x - y)$, we get $$ (x^n y^n - 1)(x - y) > (x^n - y^n)(xy - 1). $$ Or $$ (x^{n+1} - 1) (y^n - y) > (y^{n+1} - 1) (x^n - x). $$ which is the required result after dividing both sides by $(y^n -y)(x^n -x)$.