Let 0 < y_{1}< x_{1}and set

x_{n+1}= (1/2)(x_{n}+ y_{n}) and y_{n+1}= sqrt(x_{n}y_{n})

Prove that 0 < x_{n+1}- y_{n+1}< (x_{1}- y_{1})/2^{n}for n inNProvided solution:

x_{n+1}- y_{n+1 }= (1/2)(x_{n}+ y_{n}) - sqrt(x_{n}y_{n}) < (1/2)(x_{n}+ y_{n}) - y = (1/2)(x_{n}- y_{n})

Hence by induction and by the fact that 0 < y_{n}< x_{n}for n inN,0 < x_{n+1}- y_{n+1}< (x_{1}- y_{1})/2^{n }

What I do not understand is the last part of this explanation. I understand that, by induction, x_{n+1}- y_{n+1}< (x_{1}- y_{1})/2, but why 2^{n}?

Thanks in advance.