The expression $A$ is defined as a product of multiple fractions, each containing terms involving integers and the square root of 2. The simplification of $A$ leads to the form $A = P + Q\sqrt{2}$, where $P$ and $Q$ are coefficients that need to be determined. The discussion emphasizes the importance of simplifying each fraction in the product before combining them to find the final values of $P$ and $Q$. The correct simplification process is crucial for accurately determining these coefficients.
PREREQUISITESStudents, educators, and anyone interested in advanced algebraic techniques, particularly those focused on simplifying expressions involving irrational numbers and products of fractions.
your answer is correct but not simlified yetgreg1313 said:$$A=\prod_{n=1}^{2015}\dfrac{n(n+3)+\sqrt2}{(n+1)^2-2}=\prod_{n=1}^{2015}\dfrac{n^2+3n+\sqrt2}{n^2+2n-1}$$
$$=\prod_{n=1}^{2015}\dfrac{(n-(-1-\sqrt2))(n-(-2+\sqrt2)}{(n-(-1-\sqrt2))(n-(-1+\sqrt2))}=\prod_{n=1}^{2015}\dfrac{n+2-\sqrt2}{n+1-\sqrt2}$$
$$=\dfrac{3-\sqrt2}{2-\sqrt2}\cdot\dfrac{4-\sqrt2}{3-\sqrt2}\cdot\dfrac{5-\sqrt2}{4-\sqrt2}\dots=\dfrac{2017-\sqrt2}{2-\sqrt2}$$