#### \( rac{\sqrt{55}}{4}\)Question: Let $ f(u) = u - rac{u^4}{4} $ for every real number $ u $. If $ b_1 = 2 $ and $ b_{n+1} = f(b_n) $ for $ n \geq 1 $, find $ b_3 $.
["Understanding the Recursive Sequence: $ f(u) = u - \frac{u^4}{4} $, with $ b_1 = 2 $", "In this SEO-optimized article, we explore the behavior and computation of the sequence defined by the function $ f(u) = u - \frac{u^4}{4} $, starting from $ b_1 = 2 $ and recursively defined by $ b_{n+1} = f(b_n) $. We focus particularly on computing $ b_3 $, the third term in the sequence, and analyze the function-driven convergence.", "### What is the Function $ f(u) $?", "The function $ f(u) = u - \frac{u^4}{4} $ is a decreasing nonlinear transformation that models iterative correction — common in numerical algorithms and calculus-based approximations. It subtracts a quartic term from $ u $, which grows rapidly, making it useful for damping growth or modeling saturation effects.", "### Step 1: Compute $ b_2 = f(b_1) $", "Given $ b_1 = 2 $, compute:", "[\nb_2 = f(2) = 2 - \frac{2^4}{4} = 2 - \frac{16}{4} = 2 - 4 = -2\n]", "So, $ b_2 = -2 $.", "### Step 2: Compute $ b_3 = f(b_2) = f(-2) $", "Now apply $ f $ to $ b_2 = -2 $:", "[\nb_3 = f(-2) = -2 - \frac{(-2)^4}{4} = -2 - \frac{16}{4} = -2 - 4 = -6\n]", "Thus, $ b_3 = -6 $.", "### Why This Sequence Matters", "This recursive pattern illustrates how small initial perturbations can rapidly evolve under nonlinear functions. Sequence $ b_n $ begins at a positive value but quickly diverges in magnitude due to the steep negative feedback in $ -\frac{u^4}{4} $. Understanding such dynamics helps in analyzing stability in iterative processes, optimization algorithms, and even modeling damping in physics.", "### Final Answer", "After step-by-step evaluation:", "[\nb_3 = f(f(2)) = f(-2) = -6\n]", "[\n\boxed{-6}\n]", "---", "Keywords: $ f(u) = u - \frac{u^4}{4} $, recursive sequence $ b_1 = 2 $, $ b_{n+1} = f(b_n) $, compute $ b_3 $, iterative computation, nonlinear function dynamics.", "Meta Description:\nExplore how $ b_3 $ is computed in the sequence defined by $ f(u) = u - \frac{u^4}{4} $, starting with $ b_1 = 2 $. Step-by-step calculation yields $ b_3 = -6 $, illustrating rapid divergence under nonlinear feedback. Learn about fixed points and stabilization in iterative systems.", "Target Audience: Students, educators, and professionals exploring sequences, recursion, and nonlinear functions in mathematics, computer science, and applied fields."]
