Sets, Relations & Functions
Grade 11
Question:
<p>Let \(f(x) = \dfrac{x}{1-x}\). If \(x_0=\alpha,\ x_1=f(x_0),\ x_2=f(x_1),\ldots\) and \(x_{2011} = -\dfrac{1}{2012}\), find \(\alpha\).</p>
Step-by-Step Solution
<div class="solution"><p><strong>Key Idea:</strong> Reciprocal substitution converts the recurrence into an AP.</p><p><strong>Step 1:</strong> Let $y_n = 1/x_n$. Then $y_{n+1} = y_n - 1$ (arithmetic progression).</p><p><strong>Step 2:</strong> <span class="math-block">$$x_n = \frac{\alpha}{1 - n\alpha}$$</p><p><strong>Step 3:</strong> Set $x_{2011} = -1/2012$: <span class="math-block">$$\frac{\alpha}{1-2011\alpha} = -\frac{1}{2012}$$Cross-multiplying: $2012\alpha = -1+2011\alpha \implies \alpha = -1$</p><p><strong>Answer: $\alpha = -1$</strong></p><div class="trap-box"><strong>Trap:</strong> The sign matters completely. Missing the negative sign changes the answer to a non-integer value.<div class="key-concept"><strong>Key Concept:</strong> Fractional linear recurrence \to reciprocal substitution \to AP
Correct Answer: -1