# If 2*sqrt(3(2/4)*(3/5)*(4/6)...(x/y))=1 then (x,y)=?

Jul 20, 2018

$2 \cdot \sqrt{3 \left(\frac{2}{4}\right) \cdot \left(\frac{3}{5}\right) \cdot \left(\frac{4}{6}\right) \ldots \left(\frac{x}{y}\right)} = 1$

By inspection it is obvious that $y = x + 2$

$\implies 2 \cdot \sqrt{3 \left(\frac{2}{4}\right) \cdot \left(\frac{3}{5}\right) \cdot \left(\frac{4}{6}\right) \ldots \left(\frac{x}{x + 2}\right)} = 1$

$\implies 2 \cdot \sqrt{3 \cdot \frac{2 \cdot 3}{\left(x + 1\right) \left(x + 2\right)}} = 1$

Squaring both sides we get

$\implies \frac{4 \cdot 3 \cdot 2 \cdot 3}{\left(x + 1\right) \left(x + 2\right)} = 1$

$\implies {x}^{2} + 3 x + 2 - 72 = 0$

$\implies {x}^{2} + 3 x - 70 = 0$

$\implies {x}^{2} + 10 x - 7 x - 70 = 0$

$\implies x \left(x + 10\right) - 7 \left(x + 10\right) = 0$

$\implies \left(x + 10\right) \left(x - 7\right) = 0$

X can't be negative.

Hence $x = 7 \mathmr{and} y = x + 2 = 9$