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Home » The Ultimate Guide to Unlocking the Mysterious Trove You’ve Been Searching For - Leaselab

The Ultimate Guide to Unlocking the Mysterious Trove You’ve Been Searching For - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Question: A clean energy startup models the efficiency of a solar panel using the function $ E(u) = u - \frac{u^4}{4} $, where $ u $ represents normalized sunlight intensity. If $ b_n $ is defined by $ b_1 = 1 $ and $ b_{n+1} = E(b_n) $, find the value of $ b_4 $.

📌 Solution: Start with $ b_1 = 1 $ and compute successive terms using $ E(u) = u - \frac{u^4}{4} $.

📌 $ b_2 = E(b_1) = 1 - \frac{1^4}{4} = 1 - \frac{1}{4} = \frac{3}{4} $

📌 $ b_3 = E\left(\frac{3}{4}\right) = \frac{3}{4} - \frac{\left(\frac{3}{4}\right)^4}{4} = \frac{3}{4} - \frac{81}{1024} = \frac{768 - 81}{1024} = \frac{687}{1024} $

📌 $ b_4 = E\left(\frac{687}{1024}\right) = \frac{687}{1024} - \frac{\left(\frac{687}{1024}\right)^4}{4} $

📌 We compute the fourth power approximately:

📌 \left(\frac{687}{1024}\right)^4 \approx \left(0.6709\right)^4 \approx 0.205

📌 \frac{0.205}{4} \approx 0.05125 \Rightarrow b_4 \approx 0.6709 - 0.05125 = 0.61965

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