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Limit of a sequence involving square roots

Source: Indian TST Day 1 Problem 3

July 11, 2014
geometryinvariantalgebra unsolvedalgebra

Problem Statement

Starting with the triple (10072,20142,100714)(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14}), define a sequence of triples (xn,yn,zn)(x_{n},y_{n},z_{n}) by xn+1=xn(yn+zn−xn)x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})} yn+1=yn(zn+xn−yn)y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})} zn+1=zn(xn+yn−zn) z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})} for n≥0n\geq 0.Show that each of the sequences ⟨xn⟩n≥0,⟨yn⟩n≥0,⟨zn⟩n≥0\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0} converges to a limit and find these limits.
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