Let The Telescoping Dance

Algebra Level 4

\[\sqrt {1 + \frac{1}{{{1^2}}} + \frac{1}{{{2^2}}}} + \sqrt {1 + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}}} + \sqrt {1 + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}}} + ... + \sqrt {1 + \frac{1}{{{{2013}^2}}} + \frac{1}{{{{2014}^2}}}} = \frac{{ac}}{b}\]

If three consecutive natural numbers \(a\), \(b\) and \(c\) satisfy the equation above, what is \(a+b+c\)?

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