Školjka

%V0 Prove the following inequality: $$\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,$$ where $x_i > 0,$ $k \in \mathbb{N}, n \in \mathbb{N}.$