## Σάββατο, 9 Σεπτεμβρίου 2017

### 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 :

#### 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.