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📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 f(2) = 2^3 = 8.

📌 Even though the problem states "cubic polynomial," and $ x^3 $ is cubic with integer coefficients, this fits. Therefore, the answer is:

📌 Question: In a digital learning platform, student engagement $ E(t) $ over time $ t $ (in weeks) during a physics module is modeled by a function satisfying $ E(t+1) - E(t) = 2t + 1 $ for all integers $ t $, with $ E(0) = 0 $. Find $ E(10) $.

📌 E(t+1) - E(t) = 2t + 1, \quad E(0) = 0.

📌 This is a first-order difference equation. The difference $ E(t+1) - E(t) $ is linear in $ t $, so $ E(t) $ must be a quadratic polynomial. Let $ E(t) = at^2 + bt + c $.

📌 From the initial condition:

📌 E(0) = c = 0.

📌 Now compute the difference:

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