Question

Let $f\left(x\right)=\frac{x}{1-x}$ and ${}^{\prime }{a}^{\prime }$ be a real number, if $x_0=a,x_1=f\left(x_0\right),x_2=f\left(x_1\right),x_3=f\left(x_2\right)...$
If $x_{2009}=1$, then the value of ${}^{\prime }{a}^{\prime }$ is

• A. $0$
• B. $\frac{2009}{2010}$
• C. $\frac{1}{2009}$
• D. $\frac{1}{2010}$

Answer 1

The correct option is D $\frac{1}{2010}$

$x_1=f\left(x_0\right)=\frac{a}{1-a}$
$x_2=f\left(x_1\right)=\frac{x_1}{1-x_1}=\frac{a}{1-2a}$
$x_3=f\left(x_2\right)=\frac{x_2}{1-x_2}=\frac{a}{1-3a}...$
$x_{2009}=f\left(x_{2009}\right)=\frac{x_{2009}}{1-x_{2009}}=\frac{a}{1-2009a}=1$
$a=\frac{1}{2010}$