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📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 2a = 2 \Rightarrow a = 1, \quad a + b = 1 \Rightarrow 1 + b = 1 \Rightarrow b = 0.

📌 Thus, $ E(t) = t^2 $.

📌 Then $ E(10) = 10^2 = 100 $.

📌 Alternatively, recognize the recurrence as the discrete analog of derivative; sum telescoping:

📌 E(1) = E(0) + 1 = 1, \quad E(2) = 1 + (2\cdot1 + 1) = 1 + 3 = 4, \quad \text{etc.}

📌 So $ E(n) = \sum_{k=0}^{n-1} (2k + 1) = n^2 $, since sum of first $ n $ odd numbers is $ n^2 $.

📌 Hence, $ E(10) = 10^2 = 100 $.

📌 Question: An oceanographer models the depth $ d(t) $ (in meters) of a marine layer influenced by tidal forces as a cubic polynomial in time $ t $ (in days). It is known that $ d(1) = 1 $, $ d(2) = 8 $, $ d(3) = 27 $, and $ d(4) = 64 $, and $ d(t) $ achieves its minimum depth exactly once. Find $ d(0) $.

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