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2016-2007-2018-2009-2020-2011-2022-




Question Number 70044 by naka3546 last updated on 30/Sep/19
(√(2016 + 2007(√(2018 + 2009(√(2020 + 2011(√(2022 + …))))))))  =  ...
$$\sqrt{\mathrm{2016}\:+\:\mathrm{2007}\sqrt{\mathrm{2018}\:+\:\mathrm{2009}\sqrt{\mathrm{2020}\:+\:\mathrm{2011}\sqrt{\mathrm{2022}\:+\:\ldots}}}}\:\:=\:\:… \\ $$
Commented by Prithwish sen last updated on 30/Sep/19
(a−6)=(√((a−6)^2 ))=(√(a+(a^2 −13a+36)))    = (√(a+(a−9)(a−4))) = (√(a+(a−9)(√((a−4)^2 ))))  =(√(a+(a−9)(√((a+2)+(a−7)(a−2)))))  =(√(a+(a−9)(√((a+2)+(a−7)(√((a−2)^2 ))))))  =(√(a+(a−9)(√((a+2)+(a−7)(√((a+4)+(a−5)a))))))  =(√(a+(a−9)(√((a+2)+(a−7)(√((a+4)+(a−5)(√((a+6)+(a−3)(a+2)))))))))  ..........  putting a=2016 we get  (√(2016+2007(√(2018+2009(√(2020+2011(√(2022+2013....))))))))  = 2010
$$\left(\mathrm{a}−\mathrm{6}\right)=\sqrt{\left(\mathrm{a}−\mathrm{6}\right)^{\mathrm{2}} }=\sqrt{\mathrm{a}+\left(\mathrm{a}^{\mathrm{2}} −\mathrm{13a}+\mathrm{36}\right)} \\ $$$$\:\:=\:\sqrt{\mathrm{a}+\left(\mathrm{a}−\mathrm{9}\right)\left(\mathrm{a}−\mathrm{4}\right)}\:=\:\sqrt{\mathrm{a}+\left(\mathrm{a}−\mathrm{9}\right)\sqrt{\left(\mathrm{a}−\mathrm{4}\right)^{\mathrm{2}} }} \\ $$$$=\sqrt{\mathrm{a}+\left(\mathrm{a}−\mathrm{9}\right)\sqrt{\left(\mathrm{a}+\mathrm{2}\right)+\left(\mathrm{a}−\mathrm{7}\right)\left(\mathrm{a}−\mathrm{2}\right)}} \\ $$$$=\sqrt{\mathrm{a}+\left(\mathrm{a}−\mathrm{9}\right)\sqrt{\left(\mathrm{a}+\mathrm{2}\right)+\left(\mathrm{a}−\mathrm{7}\right)\sqrt{\left(\mathrm{a}−\mathrm{2}\right)^{\mathrm{2}} }}} \\ $$$$=\sqrt{\mathrm{a}+\left(\mathrm{a}−\mathrm{9}\right)\sqrt{\left(\mathrm{a}+\mathrm{2}\right)+\left(\mathrm{a}−\mathrm{7}\right)\sqrt{\left(\mathrm{a}+\mathrm{4}\right)+\left(\mathrm{a}−\mathrm{5}\right)\mathrm{a}}}} \\ $$$$=\sqrt{\mathrm{a}+\left(\mathrm{a}−\mathrm{9}\right)\sqrt{\left(\mathrm{a}+\mathrm{2}\right)+\left(\mathrm{a}−\mathrm{7}\right)\sqrt{\left(\mathrm{a}+\mathrm{4}\right)+\left(\mathrm{a}−\mathrm{5}\right)\sqrt{\left(\mathrm{a}+\mathrm{6}\right)+\left(\mathrm{a}−\mathrm{3}\right)\left(\mathrm{a}+\mathrm{2}\right)}}}} \\ $$$$………. \\ $$$$\boldsymbol{\mathrm{putting}}\:\boldsymbol{\mathrm{a}}=\mathrm{2016}\:\boldsymbol{\mathrm{we}}\:\boldsymbol{\mathrm{get}} \\ $$$$\sqrt{\mathrm{2016}+\mathrm{2007}\sqrt{\mathrm{2018}+\mathrm{2009}\sqrt{\mathrm{2020}+\mathrm{2011}\sqrt{\mathrm{2022}+\mathrm{2013}….}}}} \\ $$$$=\:\mathrm{2010}\: \\ $$

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