Unorthodox Inequality

Let $$n$$ be a positive integer and $$x$$ be a positive real. Prove that $\dfrac{n(nx+1)}{x+1} \le \dfrac{(x+1)^{2n}-1}{(x+1)^2-1}$ and find equality case.

Hint (zoom in to read):

$$\tiny _{\text{The inequality is still true even when loosening the restriction of }x\text{ to }x > -1}$$

Note by Daniel Liu
3 years, 11 months ago

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x>0 not x>=0 because if x=0 you get a non defined operation: 0/0 Nice problem ;)

- 3 years, 11 months ago

True. However, if you looked at the first hint, it can even be looser.

Has anyone solved this problem yet? I have a pretty big hint as my second hint...

- 3 years, 11 months ago

Nah I ended up at $$1 \geq 2$$ and exploded the universe.

- 3 years, 7 months ago

Final hint: multiply $$x+1$$ on both sides. Does the format of the resulting product resemble a known formula?

- 3 years, 10 months ago