Phew this one was hard to think of:
(I initially tried induction for an hour or so but got nowhere):
Credits: Assistance from Matthew LB
Solution:
So we are given the inequality:
\frac{x^{n+1}-1}{x(x^{n-1}-1)} >\frac{y^{n+1}-1}{y(y^{n-1}-1)}
Knowing that both x and y are greater than 1 we can multiply both sides by the denominator without fear of having to flip the inequality sign.
After doing so, we can move the terms around and get a different inequality:
\frac{(x^ny^n-1)}{(xy-1)} > \frac{(x^n-y^n)}{(x-y)}
Now this simplifies to:
\sum_{k=0}^{n-1} x^ky^k> \sum_{k=0}^{n-1} x^ky^{(n-1)-k}
This is true by the rearrangement inequality and we are done.
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Very nice!! 