Let The Telescoping Dance

Algebra Level 4

1+112+122+1+122+132+1+132+142+...+1+120132+120142=acb\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 aa, bb and cc satisfy the equation above, what is a+b+ca+b+c?

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