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So I have narrowed down a possible aproach to this by using the Double Angle and Half Angle Formulas, but whenever I try to apply them I get a wrong answer.

 

Here is the problem:

If \(a_0 = \sin^2 \left( \frac{\pi}{45} \right)\) and \(a_{n + 1} = 4a_n (1 - a_n)\)
for \(n \ge 0\) find the smallest positive integer \(n\) such that \(a_n = a_0\).

 Jul 23, 2020
 #1
avatar+30678 
+1

Are you sure you don't mean \(\sin^2(\frac{\pi}{5})\) ?

 Jul 24, 2020
 #3
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No, I ment \(\frac{\pi}{45}\)

That is what has been throwing me off.

Guest Jul 24, 2020
 #4
avatar+30678 
+1

Hmm, surprisingly, it does work for pi/45:

Alan  Jul 24, 2020
 #5
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Huh, thats a nice way to do it.

Thanks Alan!

Guest Jul 25, 2020
 #2
avatar+30678 
+1

I'm going to assume you mean the angle to be \(\frac{\pi}{5}\).

 

 Jul 24, 2020

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