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 \(a+b\).