Question-69296

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Question Number 69296 by ozodbek last updated on 22/Sep/19

Commented by Prithwish sen last updated on 22/Sep/19

$$\mathrm{can}\:\mathrm{you}\:\mathrm{post}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{please}? \\ $$

Commented by MJS last updated on 22/Sep/19

$$\mathrm{approximation}\:\mathrm{gives}\:\mathrm{8} \\ $$

Commented by Prithwish sen last updated on 22/Sep/19

8=3!+2!=(√((3!+2!)^2 )) = (√(2!(2!+2.3!+3.3!))) =(√(2!{2!+2.3!+3!(4−1)})) = (√(2!{2!+3!+4!))) =(√(2!{2!+(√((3!+4!)^2 ))))=(√(2!^2 +2!(√(3!(3!+2.4!+4.4!))))) =(√(2!^2 +2!(√(3!{3!+2.4!+4!(5−1)})))) =(√(2!^2 +2!(√(3!{3!+(4!+5!))))) =(√(2!^2 +2!(√(3!^2 +3!(√((4!+5!)^2 )))))) ...... =(√(2!^2 +2!(√(3!^2 +3!(√(4!^2 +4!(√(.......)))))))) Thanks Sir MJS your approximation help me a lot. I appreciate your cooperation. Again thanks a lot.

$$\mathrm{8}=\mathrm{3}!+\mathrm{2}!=\sqrt{\left(\mathrm{3}!+\mathrm{2}!\right)^{\mathrm{2}} \:}\:=\:\sqrt{\mathrm{2}!\left(\mathrm{2}!+\mathrm{2}.\mathrm{3}!+\mathrm{3}.\mathrm{3}!\right)} \\ $$$$=\sqrt{\mathrm{2}!\left\{\mathrm{2}!+\mathrm{2}.\mathrm{3}!+\mathrm{3}!\left(\mathrm{4}−\mathrm{1}\right)\right\}}\:=\:\sqrt{\mathrm{2}!\left\{\mathrm{2}!+\mathrm{3}!+\mathrm{4}!\right)} \\ $$$$=\sqrt{\mathrm{2}!\left\{\mathrm{2}!+\sqrt{\left(\mathrm{3}!+\mathrm{4}!\right)^{\mathrm{2}} }\right.}=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!\left(\mathrm{3}!+\mathrm{2}.\mathrm{4}!+\mathrm{4}.\mathrm{4}!\right)}} \\ $$$$=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!\left\{\mathrm{3}!+\mathrm{2}.\mathrm{4}!+\mathrm{4}!\left(\mathrm{5}−\mathrm{1}\right)\right\}}}\: \\ $$$$=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!\left\{\mathrm{3}!+\left(\mathrm{4}!+\mathrm{5}!\right)\right.}}\: \\ $$$$=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!^{\mathrm{2}} +\mathrm{3}!\sqrt{\left(\mathrm{4}!+\mathrm{5}!\right)^{\mathrm{2}} }}}\:\:…… \\ $$$$=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!^{\mathrm{2}} +\mathrm{3}!\sqrt{\mathrm{4}!^{\mathrm{2}} +\mathrm{4}!\sqrt{…….}}}} \\ $$$$\boldsymbol{\mathrm{Thanks}}\:\boldsymbol{\mathrm{Sir}}\:\boldsymbol{\mathrm{MJS}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{approximation}}\:\boldsymbol{\mathrm{help}}\:\boldsymbol{\mathrm{me}} \\ $$$$\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{lot}}.\:\boldsymbol{\mathrm{I}}\:\boldsymbol{\mathrm{appreciate}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{cooperation}}.\:\boldsymbol{\mathrm{Again}} \\ $$$$\boldsymbol{\mathrm{thanks}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{lot}}. \\ $$

Commented by MJS last updated on 22/Sep/19