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Ayaan Barr 2022-07-07 Answered
Let A 0 = 3 4 , and A n + 1 = 1 + A n 2 for all n 0
How to find the value of n = 1 A n ?
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Answers (1)

cefflid6y
Answered 2022-07-08 Author has 13 answers
If we set A n = cos 2 ( θ n ), we get:
A n + 1 = 1 + cos ( θ n ) 2 = cos 2 ( θ n 2 ) ,
and since θ 0 = π 6 , induction gives:
A n = cos 2 ( π 6 2 n ) = ( sin ( π 6 2 n 1 ) 2 sin ( π 6 2 n ) ) 2
from which it follows that:
n = 1 + A n = ( 3 π ) 2 = 9 π 2 .

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