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Home » \( a=1, b=0 \): 3 - DNSFLEX

\( a=1, b=0 \): 3 - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 #### 157.63Question: A climatologist models the rate of ice melt $ M(t) $ at time $ t $ (in years) as $ M(t) = t - \frac{t^3}{3} $. If $ b_n $ is defined recursively by $ b_1 = 1 $ and $ b_{n+1} = M(b_n) $, compute $ b_4 $.

📌 Solution: We are given $ M(t) = t - \frac{t^3}{3} $ and $ b_1 = 1 $. Compute successive terms:

📌 $ b_1 = 1 $

📌 $ b_2 = M(b_1) = 1 - \frac{1^3}{3} = 1 - \frac{1}{3} = \frac{2}{3} $

📌 $ b_3 = M(b_2) = \frac{2}{3} - \frac{(\frac{2}{3})^3}{3} = \frac{2}{3} - \frac{8}{81} = \frac{54 - 8}{81} = \frac{46}{81} $

📌 $ b_4 = M(b_3) = \frac{46}{81} - \frac{(\frac{46}{81})^3}{3} $

📌 First, compute $ \left( \frac{46}{81} \right)^3 = \frac{97336}{531441} $, so

📌 b_4 = \frac{46}{81} - \frac{1}{3} \cdot \frac{97336}{531441} = \frac{46}{81} - \frac{97336}{1594323}

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