1. a) Verify the identity $\sqrt{1+\frac{1}{n^{2}}+\frac{1}{(n+1)^{2}}}=1+\frac{1}{n}-\frac{1}{n+1}, \forall n \in \mathbb{N}^{*}$.

b) Determine the integer part of the number

$$
a=\sqrt{1+\frac{1}{1^{2}}+\frac{1}{2^{2}}}+\sqrt{1+\frac{1}{2^{2}}+\frac{1}{3^{2}}}+\ldots+\sqrt{1+\frac{1}{2012^{2}}+\frac{1}{2013^{2}}}
$$
1. a) Square both sides.

b) Using point a) and the telescoping sum procedure, we obtain that $a=2013-\frac{1}{2013}$, so $\lfloor a\rfloor=2012$.