Let $ x_1, x_2, \ldots, x_n \; \in \; \mathbb{R}_{+} $ so that $ x_1 \geq x_2 \geq \cdots \geq x_n $
We also know that: $$ \frac{1}{2^{x_1}} + \frac{1}{2^{x_2}} + \cdots + \frac{1}{2^{x_n}} = 1 $$
Prove the following inequality: $$ \frac{1}{3^{x_1}} + \frac{2}{3^{x_2}} + \cdots + \frac{n}{3^{x_n}} \geq \frac{n + 1}{2n} $$
I've tried to prove it several times but I haven't succeeded. I would really appreciate if you can help me.