Euler's Method for Differential Equations

Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by $$a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),$$ where there are $n$ iterations of $L$. For example, $$a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).$$ As $n$ approaches infinity, what value does $n a_n$ approach? Express your answer as a fraction p/q in simplest terms.

Source: Math Prize for Girls