The reason the only solution is x=4 for all values of n is as follows:
If f_n(x)=2x for all n then f_n+1(x)=2x since n+1 is just another value.
This means that f_n+1(x)=sqrt(x^2+6f_n(x)) can be reduced to 2x=sqrt(x^2+6*2x). This is solved below:
Therefore x = 0 or 4
However x=0 is not a solution since for n=0 f_0(x)=2*0=0 but we have been given that f_0(x)=8
x=4 does of course work for n=0 and is therefore the only solution for all n.
Posted by Paul
on 2006-11-20 10:07:25