We've got to find the minimum value of n for which $$ \prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{\sqrt{27k^3+54k^2+36k+8}}\right)}{\arctan \left( \frac{1}{\sqrt{3k+1}} \right)}>2000$$ So I just did some simplifications hoping I'd see something: $$\prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{(3k+2)^{\frac{3}{2}}}\right)}{\arcsin \left( \frac{1}{\sqrt{3k+2}} \right)}>2000$$ I'm thinking that maybe it's something like telescopic series and am trying to bring it into those kinds of terms. Or can it be something else? \
Edit: I think I got it the numerator is just 3 times the denominator isn't it?