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Let f(x)=x/(1-x), where a is a real number. Define the sequence by x0=a, x1=f(x0), x2=f(x1), x3=f(x2), and so on. If x2009=1, then what is the value of a?

1. 0
2. 2009/2010
3. 1/2009
4. 1/2010

Correct answer: 1/2010

Solution

With f(x)=x/(1-x), iterating gives x_n = a/(1 - n*a). Setting x_2009 = a/(1 - 2009a) = 1 gives a = 1 - 2009a, so 2010a = 1 and a = 1/2010.

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