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Sum in a recursive series in a sum

Source: Poland National Olympiad 1992 First Round #6

June 2, 2023

Problem Statement

The sequence (xn)(x_n) is determined by the conditions: x0=1992,xn=−1992n⋅∑k=0n−1xkx_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k for n≥1n \geq 1. Find ∑n=019922nxn\sum_{n=0}^{1992} 2^nx_n.
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