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📅 May 9, 2026 👤 scraface

May 09, 2026

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📌 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:

📌 E(t+1) - E(t) = a(t+1)^2 + b(t+1) - (at^2 + bt) = a(2t+1) + b = 2at + (a + b).

📌 Set equal to $ 2t + 1 $:

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