Sets, Relations & Functions Question - None

Question

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 $\alpha$ .

Accepted Answer

Key Idea: Reciprocal substitution converts the recurrence into an AP.

Step 1: Let $$y_n = 1/x_n$$ . Then $y_{n+1} = y_n - 1$ (arithmetic progression).

Step 2: $x_n = \frac{\alpha}{1 - n\alpha}$

Step 3: Set $x_{2011} = -1/2012$ : $\frac{\alpha}{1-2011\alpha} = -\frac{1}{2012}$ Cross-multiplying: $2012\alpha = -1+2011\alpha \implies \alpha = -1$

Answer: $\alpha = -1$

Trap: The sign matters completely. Missing the negative sign changes the answer to a non-integer value.

Key Concept: Fractional linear recurrence → reciprocal substitution → AP

Question:

Step-by-Step Solution

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