British Maths Olympiad 2015: Can You Find The 2015th Term?

Given that the first term x_1 of a sequence is 2014, the sequence satisfies the following iterative formula which is x_n+1 = [(√2+1)x_n−1]/[(√2+1)+x_n], now, I want you to find the 2015th term, x2015. Can you find it? Give this problem a try and this problem comes from the 2015 British Maths Olympiad Round 2.
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[@AxAllin]

We can just calculate x(n+2) and then x(n+4) and x(n+8) =xn. We get that our sequence is periodic and the rest is easy

[@1psi3colourmath]

Yeah, it's doable that way, just that it may be messy which I think using trigonometry is the finest way to do it :)

[@parthosaha4170]

The tan22.5 substitution and the x_n=tana_n substitution killed the problem

[@1psi3colourmath]

Indeed, the tan substitution is a powerful technique in solving sequence like this.

[@NaN_000]

This one is hard for me.
Btw I like the Tangent method 🙂

[@1psi3colourmath]

Glad you liked it!