I have been trying to solve the following equation without any success:
$$(n+1)X^{n+1}-(n-1)X^{n}-(n-1)X + n+1 =0$$
I already tried to use the known formula: $$x^{n+1} + \frac{1}{x^{n+1}} = \left( x + \frac{1}{x} \right)\left( x^{n} + \frac{1}{x^{n}} \right) - \left( x^{n-1} + \frac{1}{x^{n-1}} \right)$$
in: $$(n+1)(X^{n+1} + X^{-(n+1)}) -(n-1)(X^{n} + X^{-n} + X+ X^{-1})+ 2(n+1) = 0$$ but I was not able to solve it. Any help or hint would be appreciated.