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 \( b_3 \).

["Understanding the Function ( M(u) = u - \frac{u^4}{4} ) and Computing the Sequence ( b_n )", "The mathematical function ( M(u) = u - \frac{u^4}{4} ) is a quartic transformation that plays a key role in modeling nonlinear systems with damping or correction effects. Defined for all real ( u ), this function exhibits behavior useful in fields such as approximation theory, dynamical systems, and iterative algorithms. In this article, we explore the function’s properties and compute the third term, ( b_3 ), in a recursively defined sequence based on ( M ).", "### What is ( M(u) = u - \frac{u^4}{4} )?", "The function ( M(u) = u - \frac{u^4}{4} ) combines a linear term ( u ) with a negative quartic correction ( -\frac{u^4}{4} ). For small values of ( u ), the linear term dominates, but as ( |u| ) increases, the quartic term becomes significant, slowing or reversing growth. This makes ( M(u) ) useful for modeling systems where growth is naturally limited—such as certain population models, physical damping systems, or mathematical approximations requiring nonlinear correction.", "### The Recursive Sequence ( b_n )", "We are given a recursive sequence defined by:\n- ( b_1 = 1 )\n- ( b_n = M(b_{n-1}) ) for ( n \geq 2 )", "This means each term is generated by applying the function ( M ) to the previous term.", "### Computing ( b_3 )", "We compute the first three terms step by step.", "Step 1: Compute ( b_2 )", "Using ( b_1 = 1 ):", "[\nb_2 = M(b_1) = M(1) = 1 - \frac{1^4}{4} = 1 - \frac{1}{4} = \frac{3}{4}\n]", "Step 2: Compute ( b_3 )", "Now use ( b_2 = \frac{3}{4} ):", "[\nb_3 = M\left( \frac{3}{4} \right) = \frac{3}{4} - \frac{\left( \frac{3}{4} \right)^4}{4}\n]", "First, compute ( \left( \frac{3}{4} \right)^4 ):", "[\n\left( \frac{3}{4} \right)^4 = \frac{81}{256}\n]", "Now divide by 4:", "[\n\frac{81}{256} \div 4 = \frac{81}{256} \cdot \frac{1}{4} = \frac{81}{1024}\n]", "Now compute ( b_3 ):", "[\nb_3 = \frac{3}{4} - \frac{81}{1024} = \frac{768}{1024} - \frac{81}{1024} = \frac{687}{1024}\n]", "Thus,", "[\nb_3 = \frac{687}{1024}\n]", "### Summary and Final Answer", "The function ( M(u) = u - \frac{u^4}{4} ) is a smoothly decreasing nonlinear map for ( u > 0 ), useful in modeling bounded dynamical systems. Through recursion with ( b_1 = 1 ), we computed:", "[\nb_2 = \frac{3}{4}, \quad b_3 = \frac{687}{1024}\n]", "So, the value of ( b_3 ) is ( \frac{687}{1024} ).", "Understanding such recursive sequences helps in numerical analysis, algorithmic design, and the study of nonlinear recurrence relations—key topics in applied mathematics and computational science."]