The discussion revolves around the simplification of the expression \( A \), which is defined as a product of fractions involving arithmetic operations and the square root of 2. The focus is on finding a simplified form of \( A \) in terms of two components, \( P \) and \( Q \), where \( A = P + Q\sqrt{2} \).
There is no consensus on the simplification of \( A \), as the second participant implies that further work is needed to achieve the desired form.
The discussion does not clarify the specific methods or steps required to simplify \( A \), nor does it address any assumptions or definitions that may affect the simplification process.
your answer is correct but not simlified yetgreg1313 said:$$A=\prod_{n=1}^{2015}\dfrac{n(n+3)+\sqrt2}{(n+1)^2-2}=\prod_{n=1}^{2015}\dfrac{n^2+3n+\sqrt2}{n^2+2n-1}$$
$$=\prod_{n=1}^{2015}\dfrac{(n-(-1-\sqrt2))(n-(-2+\sqrt2)}{(n-(-1-\sqrt2))(n-(-1+\sqrt2))}=\prod_{n=1}^{2015}\dfrac{n+2-\sqrt2}{n+1-\sqrt2}$$
$$=\dfrac{3-\sqrt2}{2-\sqrt2}\cdot\dfrac{4-\sqrt2}{3-\sqrt2}\cdot\dfrac{5-\sqrt2}{4-\sqrt2}\dots=\dfrac{2017-\sqrt2}{2-\sqrt2}$$