Recurring Styles #14 - Nested Radicals!

Calculus Level 5

\[\large{ x_{n+1} = \sqrt{2+x_n} \quad, \quad x_1 = \sqrt{2} }\]

Consider the sequence \((x_n)_{n \in \mathbb Z^+} \) as defined above. Let \(L = \displaystyle \lim_{n \to \infty} 4^n(2-x_n) \). If \(L\) can be expressed in the form \(\dfrac{A}{B} \pi^C \) for positive integers \(A,B,C\), find the minimum value of \(A+B+C\).

Bonus: Generalize \(x_n\) in terms of \(n\).

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