TR-MO-R1 2017 Problem 22

$f(0)=\dfrac{2}{3}$ ve $n=1,2,3,\ldots$ için $f(n)\neq 0$ ve $\big(f(n+1)-1\big)\big(f(n)+3\big)+3=0$ olduğuna göre

$$\frac{1}{f(0)}+\frac{1}{f(1)}+\frac{1}{f(2)}+\frac{1}{f(3)}+\cdots+\frac{1}{f(2016)}+\frac{1}{f(2017)}=?$$

A) $3^{2018}-1010$ B) $3^{2017}-1009$ C) $2\cdot 3^{2018}-1009$ D) $2\cdot\left(3^{2017}-505\right)$ E) $2\cdot 3^{2017}-1009$