1988 AHSME Problems/Problem 30

Problem

Let $f(x) = 4x - x^{2}$ . Give $x_{0}$ , consider the sequence defined by $x_{n} = f(x_{n-1})$ for all $n \ge 1$ . For how many real numbers $x_{0}$ will the sequence $x_{0}, x_{1}, x_{2}, \ldots$ take on only a finite number of different values?

$\textbf{(A)}\ \text{0}\qquad \textbf{(B)}\ \text{1 or 2}\qquad \textbf{(C)}\ \text{3, 4, 5 or 6}\qquad \textbf{(D)}\ \text{more than 6 but finitely many}\qquad \textbf{(E) }\infty$

Solution

Apply one of the standard formulae for the gradient of the line of best fit, e.g. $\frac{\frac{\sum {x_i y_i}}{n} - \bar{x} \bar{y}}{\frac{\sum {x_{i}^2}}{n} - \bar{x}^2}$ , and substitute in the given condition $x_3 - x_2 = x_2 - x_1$ . The answer is $\boxed{\text{A}}$ .

1988 AHSME (Problems • Answer Key • Resources)
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1988 AHSME Problems/Problem 30

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