1975_Question_6

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.

:grinning:

1 Like

Very nice!! :sunglasses: :sunglasses: :sunglasses: