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Linear Polynomial

Source: 1986 National High School Mathematics League, Exam Two, Problem 1

February 24, 2020
algebrapolynomial

Problem Statement

For real numbers a0,a1,⋯ ,an(a0≠a1)a_0,a_1,\cdots,a_n(a_0\neq a_1), we haveai−1+ai+1=2aia_{i-1}+a_{i+1}=2a_i for i=1,2,⋯ ,n−1i=1,2,\cdots,n-1. Prove that P(x)=a0Cn0(1−x)n+a1Cn1x(1−x)n−1+⋯+anCnnxnP(x)=a_0\text{C}_n^0(1-x)^n+a_1\text{C}_n^1x(1-x)^{n-1}+\cdots+a_n\text{C}_n^nx^n is a linear polynomial.
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