Ta có
$A = \sqrt{1 + 2014^2 + \dfrac{2014^2}{2015^2}} + \dfrac{2014}{2015}$
$= \sqrt{\dfrac{2015^2 + 2014^2. 2015^2 + 2014^2}{2015^2}} + \dfrac{2014}{2015}$
$= \dfrac{1}{2015} \sqrt{(2014+1)^2 + 2014^2 (2014+1)^2 + 2014^2} + \dfrac{2014}{2015}$
$= \dfrac{1}{2015} \sqrt{2014^4 + 2.2014^3 + 3.2014^2 + 2.2014 + 1} + \dfrac{2014}{2015}$
$= \dfrac{1}{2015} \sqrt{2014^2 (2014^2 + 2014 + 1) + 2014(2014^2 + 2014 + 1) + (2014^2 + 2014 + 1)} + \dfrac{2014}{2015}$
$ = \dfrac{1}{2015} \sqrt{(2014^2 + 2014 + 1)^2} + \dfrac{2014}{2015}$
$ = \dfrac{2014^2 + 2014 +1}{2015}+ \dfrac{2014}{2015}$
$= \dfrac{2014^2 + 2.2014 + 1}{2015}$
$= \dfrac{(2014 + 1)^2}{2015}$
$= \dfrac{2015^2}{2015} = 2015$
