\( b_3 = M(b_2) = M\left(\frac{3}{4}\right) = \frac{3}{4} - \frac{\left(\frac{3}{4}\right)^4}{4} \)

February 22, 2026

$b_3 = M(b_2) = M\left(\frac{3}{4}\right) = \frac{3}{4} - \frac{\left(\frac{3}{4}\right)^4}{4}$

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Define \( M(u) = u - \frac{u^4}{4} \) for all real \( u \). If \( n \) is a positive integer, define \( b_n \) by \( b_1 = 1 \) and \( b_{n} = M(b_{n-1}) \) for \( n \geq 2 \). Compute \( b_3 \).
We compute recursively:
\( b_2 = M(b_1) = M(1) = 1 - \frac{1^4}{4} = 1 - \frac{1}{4} = \frac{3}{4} \)
Compute \( \left(\frac{3}{4}\right)^4 = \frac{81}{256} \), so:
b_3 = \frac{3}{4} - \frac{1}{4} \cdot \frac{81}{256} = \frac{3}{4} - \frac{81}{1024}
Convert \( \frac{3}{4} = \frac{768}{1024} \):

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