This problem was from the 2004 Luzerne County Math Contest:
If (r + 1/r)^2 = 3, what is r + (1/r^3) = ?
Any help would really be appreciated. Thanks in advance!
I don't know if there is a more elegant way of solving this but:
You can solve
which has complex roots but which can be represented in exponential form quite nicely.
Substitute into:
and simplify, and if what I have done has no algebraic errors you should end up with
(note: in a competition you will get no marks for the final answer but just from the method you used to find it)
CB