√x and F(x)=4
√x. Determine the value of each of the following expressions:
(i) C(25 + C(6 + C(5 + C(25 + C(6 + C(5+.....))))))
(ii) F(14 + F(5 + F(79 + F(14 + F(5+ F(79+ .....))))))
Can you come up with an analytic (apart from a computer program) solution?
(In reply to For the simple case
Using the trick makes this problem equivalent to solving:
(((x^3-25)^3-6)^3-5)^3-x = 0
(((x^4-15)^4-5)^4-79)^3-x = 0
If we were expected to find a rational value for each case, then this would be doable with minimal computer assistance. First we would only need the leading coefficient and constant term of each polynomial. Then a finite set of candidates would be generated by the possible fractions made by factor of the constant divided by a factor of the leading coefficient.
In both cases the leading coefficient is 1, so the only rational solutions are integer solutions.
The first equation does have an integer solution, x=3. The second does not have an integer solution. I numerically calculated 1.8918763177 as a root.