Let $n$ distinct points $i \in \mathbb{N}$ be on the interval $[1, -1]$. Define $p_{k}$ as,
$ p_k = \prod_{1 \leq j \neq k \leq n} |x_{k} - x_{j}|$
Prove that,
$ \sum_{k=1}^{n} \frac{1}{p_{k}} \geq 2^{(n - 2)} $
My work so far
$\frac{2}{p_1} \geq 1$
$\sum_{k=1}^{k} \frac{1}{p_{k}} \geq 2^{(k - 2)}$
We want to prove it for $(k + 1)$ $\sum_{k=1}^{k+1} \frac{1}{p_{k}} = \left(\sum_{k=1}^{k} \frac{1}{p_{k}}\right) + \frac{1}{p_{k+1}}$
By induction
$\sum_{k=1}^{k} \frac{1}{p_{k}} \geq 2^{(k - 2)}$
$\left(\sum_{k=1}^{k} \frac{1}{p_{k}}\right) + \frac{1}{p_{k+1}} \geq 2^{(k - 2)} + \frac{1}{p_{k+1}}$
Can someone help?