Let C(x)=
^{3}√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?
The method used in Power to the 2 http://perplexus.info/show.php?pid=1525 can work here, but due to cubes and fourth roots, the resulting polynomials would be too complex to analyze the same way. This would require some better analysis.
If we assume K chose the numbers to provide a simple answer, we can input the relevant polynomials of each equation.
In the case of ((x^325)^36)^3=x+5 we note that 3^3 is close to 25 and also 3+5 is a perfect cube. Trying x=3, we find it works out, thus implying that the quantity in (i) equals 3 as the number of cube roots approaches infinity.
A similar method in (ii) would seem to work, as 2^4 is close to 14 and 79+2 is a perfect square, however it actually doesn't  it would need to read F(14 + C(5 + F(79 + F(14 + ...) for that to be the case.

Posted by Gamer
on 20070108 00:30:50 