To see what's going on, replace the x=2014 with smaller numbers.
tan(arctan(1)+arctan(1/3))=2 [this was a recent problem solved via origami]
So there's a pattern here. The answer would seem to be N=2015
The addition of a single term seems to make the answer go up by 1. Lets see what happens when we work backwards -- try to make the answer go up by one to see what the term must be.
My calculator says tan(arctan(4)-arctan(3))=1/13
which is promising. That's 3²+3+1
tan(arctan(x+1)-arctan(x)) = [(x+1)-(x)]/[1+(x+1)(x)] = (x2+x+1)-1
So this is the term that makes the answer increase by 1.
So I've proven N=x+1
So when x=2014, N=2015.
Posted by Jer
on 2014-07-01 11:17:51