If three successive terms of a geometric sequence with ratio r correspond to the lengths of the three sides of a triangle, then determine whether or not [r]+[-r]=-1.

[x] is the greatest integer ≤ x.

Sum of two sides is greater than the 3rd side.
rē<1+r
1<rē+r
For these 2 equations to satisfy
(1-√5)/2 < r < (-1+√5)/2
Clearly -1 < r < 1
=> [r]={0,-1} and [-r]={-1,0}
=> [r]+[-r]=-1

Posted by Praneeth on 2007-12-27 03:17:25