Unexpected root of equation | |
Tower of Pisa
What is the value of : (1)
if the process continues indefinitely.
Here we define:
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Investigation
If the value of (1) is x. Then we can easily observe that:
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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
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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.
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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.
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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
ak < 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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Σάββατο 9 Σεπτεμβρίου 2017
Unexpected root of equation
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