MHB Simplify $A$: Multiply Fractions

[Albert1]

$A=(\dfrac {1\times 4+\sqrt 2}{2\times 2-2})\times (\dfrac {2\times 5+\sqrt 2}{3\times 3-2})\times(\dfrac {3\times 6+\sqrt 2}{4\times 4-2})\times --------\times (\dfrac {2015\times 2018+\sqrt 2}{{2016}\times {2016}-2})$

please simplify $A$

 

[Greg]

Spoiler

$$A=\prod_{n=1}^{2015}\dfrac{n(n+3)+\sqrt2}{(n+1)^2-2}=\prod_{n=1}^{2015}\dfrac{n^2+3n+\sqrt2}{n^2+2n-1}$$

$$=\prod_{n=1}^{2015}\dfrac{(n-(-1-\sqrt2))(n-(-2+\sqrt2)}{(n-(-1-\sqrt2))(n-(-1+\sqrt2))}=\prod_{n=1}^{2015}\dfrac{n+2-\sqrt2}{n+1-\sqrt2}$$

$$=\dfrac{3-\sqrt2}{2-\sqrt2}\cdot\dfrac{4-\sqrt2}{3-\sqrt2}\cdot\dfrac{5-\sqrt2}{4-\sqrt2}\dots=\dfrac{2017-\sqrt2}{2-\sqrt2}$$

 

[Albert1]

Spoiler

$$A=\prod_{n=1}^{2015}\dfrac{n(n+3)+\sqrt2}{(n+1)^2-2}=\prod_{n=1}^{2015}\dfrac{n^2+3n+\sqrt2}{n^2+2n-1}$$

$$=\prod_{n=1}^{2015}\dfrac{(n-(-1-\sqrt2))(n-(-2+\sqrt2)}{(n-(-1-\sqrt2))(n-(-1+\sqrt2))}=\prod_{n=1}^{2015}\dfrac{n+2-\sqrt2}{n+1-\sqrt2}$$

$$=\dfrac{3-\sqrt2}{2-\sqrt2}\cdot\dfrac{4-\sqrt2}{3-\sqrt2}\cdot\dfrac{5-\sqrt2}{4-\sqrt2}\dots=\dfrac{2017-\sqrt2}{2-\sqrt2}$$

your answer is correct but not simlified yet
$A=P+Q\sqrt 2$
$P=? ,Q=?$

 

[Greg]

Spoiler

$$P=2016,\quad\,Q=\dfrac{2015}{2}$$