I sense a pattern!

Algebra Level 2

$\dfrac{1}{2^{2}-1}+\dfrac{1}{4^{2}-1}+\dfrac{1}{6^{2}-1}+\dfrac{1}{8^{2}-1}+\cdots+\dfrac{1}{1000^{2}-1}\$

The value of the sum above can be expressed as $\frac{a}{b},$ where $a$ and $b$ are coprime positive integers. Find the value of $a+b$ .