Let The Telescoping Dance

Algebra Level 3

\[\begin{array}{l} {\rm{If}}\\ \sqrt {1 + \frac{1}{{{1^2}}} + \frac{1}{{{2^2}}}} + \sqrt {1 + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}}} + \sqrt {1 + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}}} + ... + \sqrt {1 + \frac{1}{{{{2013}^2}}} + \frac{1}{{{{2014}^2}}}} = \frac{{ac}}{b}\\ a,b,\;{\rm{and}}\;c\;{\rm{are}}\;{\rm{three}}\;{\rm{concecutive}}\;{\rm{natural}}\;{\rm{numbers}}\\ {\rm{What}}\;{\rm{is}}\;{\rm{the}}\;{\rm{value}}\,{\rm{of}}\,a + b + c? \end{array}\]

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