Σάββατο, 9 Σεπτεμβρίου 2017

Unexpected root of equation

Unexpected root of equation


Tower of Pisa


What is the value of :                                                            (1)
if the process continues indefinitely.

Here we define:
                


Investigation                

If the value of (1) is x. Then we can easily observe that:
                                    
                          or        x2 = 2x             (2)


Question         


 Solve the equation for all real roots x :
                       
(correct your answers to 1 decimal place)


Solution   


            x = 2,  4 or  –0.8 (to 1 dec. pl)
The unexpected root can be found by studying the following curve 



The unexpected root x = –0.766 664 7 …. can be found with more degree of accuracy by using Newton’s method for approximation of roots.

 

Back to Tower of Pisa problem


If we have we have three roots for (2), what then is the value of (1).
Obviously x = -0.766 664 7… cannot be the value of (1) since (1) is positive.

So the choice is narrow down to x = 2 or 4.

 

Monotone bounded theorem


If we take :
         
(1)            Bounded
         We use induction to show that an is bounded by 2.
         Obviously  a1 = 1.4142… <2
         Assume ak-1 < 2
         Then  
         \  an < 2  "nÎN

(2)            Monotonic increasing
         Obviously   a1 = 1.4142….,   a2 = 1.6325….
         \    a1 < a2.
         Assume    ak-1 < ak.
         Then       
                         
                                 a < ak+1.
                    \    an < an+1   and the sequence is monotonic increasing.

(3)            By the Bounded monotone theorem,  an has a limit.
         Since  an is bounded by 2,  the value of (1) is 2, and not 4.

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