Points Marching on a Hyperbola

★ 8/10Recursion sequences Estimation analytic geometry Casework

Problem

On the curve $C: xy = 1$ , starting from $A_1(x_1, y_1)$ with $x_1 = 1$ , each point $A_n(x_n, y_n)$ is joined to the next intersection $A_{n+1}$ of $C$ with the line through $A_n$ of slope

k_n = -\frac{x_n + 1}{x_n^2 + 4x_n}.

1. Find the relation between $x_{n+1}$ and $x_n$ . 2. Compare $x_n$ with $2$ , and prove your claim. 3. Prove that

|x_1 - 2| + |x_2 - 2| + \cdots + |x_n - 2| < 2.

Answer

Solution

Difficulty	8/10
Topics	Recursion, sequences, Estimation, analytic geometry, Casework

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