Let \(n\) be a positive integer and \(P(x)\) a polynomial of degree \(2n\) such that \(P(0) = 1\) and \(P(k) = 2^{k-1}\) for \(k=1,2,3, \ldots, 2n\).

Prove that \(2P(2n+1) - P(2n+2)=1\).

Let \(n\) be a positive integer and \(P(x)\) a polynomial of degree \(2n\) such that \(P(0) = 1\) and \(P(k) = 2^{k-1}\) for \(k=1,2,3, \ldots, 2n\).

Prove that \(2P(2n+1) - P(2n+2)=1\).

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TopNewestI found a proof with method of differences. Interesting how one can consider part of the chart to be a function following f(n)=2^n for n in domain, but then have an extra column with alternating 1s and 0s. My latex is bad. I hope you see where I am going. – Sal Gard · 1 year ago

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