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Home » Vessel Sink: Why This Sunken Ship Wasn’t What It Appeared To Be! - Sourci

Vessel Sink: Why This Sunken Ship Wasn’t What It Appeared To Be! - Sourci

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 2f(x) - x = 2(x + 2) - x = 2x + 4 - x = x + 4.

📌 So $ f(f(x)) = 2f(x) - x $ holds. Also, $ f(1) = 3 $, which matches.

📌 Thus, $ f(x) = x + 2 $ is a solution. Since the functional equation constrains its behavior and the recurrence propagates uniquely from $ f(1) = 3 $, the solution is unique under mild regularity (e.g., continuity implied by recursion), so $ f(x) = x + 2 $ for all $ x $.

📌 By the Remainder Theorem for polynomials, the remainder $ R(t) $ when dividing by a degree-2 polynomial is linear:

📌 for some polynomial $ Q(t) $. To find $ a $ and $ b $, evaluate at the roots $ t = 1 $ and $ t = 2 $.

📌 Since $ t = 1 $ makes the divisor zero, the remainder satisfies:

📌 Solution: We are given a recursive sequence:

📌 First, compute the first few terms of $ a_n $:

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