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Identity with binomial coefficients

Source: Czech and Slovak Olympiad 1983, National Round, Problem 4

April 10, 2020
algebraBinomial coefficient identityArithmetic Progressionnational olympiadarithmetic sequencebinomial coefficients

Problem Statement

Consider an arithmetic progression a0,…,ana_0,\ldots,a_n with n≥2n\ge2. Prove that ∑k=0n(−1)k(nk)ak=0.\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.
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