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Astute Root Evaluation (Posted on 2007-01-07) Difficulty: 3 of 5
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?

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts re: For the simple case Comment 2 of 2 |
(In reply to For the simple case by Gamer)

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.

  Posted by Brian Smith on 2016-12-18 15:16:16
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