Let $V = \displaystyle \large \sum_{i=1}^\infty \frac{(-1)^{i+1}}{F_i \cdot F_{i+1}}$ where $F_i$ is the $i$th Fibonacci number. If $V = \large \frac{\sqrt{a} – b}{c}$ where $a$ is a number indivisible by a square of an integer and $b$ and $c$ are relatively prime, what is $2000 \cdot a – 100 \cdot b – 5 \cdot c$? (Just as a clarification, $F_1 = 1$ and $F_2 = 1$.)