Inequality about sums and products

Question

Let $x_1,x_2,...,x_n$ be positive real numbers. Show that $$\frac {1} {2^n \times \sqrt {x_1 \times x_2 \times ... \times x_n} } + \sum_ {k=1}^n \frac {x_k} {(x_1+1)(x_2+1)...(x_k+1)} \ge 1.$$ I tried to use AM-GM, but I didn't get to any result.

Answer 1

$\frac{1}{2^n\sqrt{\prod\limits_{i=1}^n}x_i}+\sum\limits_{k=1}^n\frac{x^k}{\prod\limits_{i=1}^k(1+x_i)}\geq1\Leftrightarrow\frac{1}{2^n\sqrt{\prod\limits_{i=1}^n}x_i}+\sum\limits_{k=1}^n\frac{1+x_k}{\prod\limits_{i=1}^k(1+x_i)}\geq1+\sum\limits_{k=1}^n\frac{1}{\prod\limits_{i=1}^k(1+x_i)}\Leftrightarrow$

$\Leftrightarrow\frac{1}{2^n\sqrt{\prod\limits_{i=1}^n}x_i}\geq\frac{1}{\prod\limits_{i=1}^n(1+x_i)}$, which is just AM-GM