# Do You See The Connection?

Algebra Level 5

Let the sequence of real numbers $${x_n}$$ be defined with

$\begin{cases} x_1=k\\ x_{n+1} = 4x_n(1-x_n),\ \ n \geq 1. \end{cases}$

Let there be $$N$$ distinct values of $$k$$ such that $$x_{2014} = 0$$. Find the last three digits of $$N$$.

Note: For those interested, this question was partly inspired by the function $$f(x) = \lambda x(1-x)$$ also know as the logistic function and is related to population growth.

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