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
$$\mathrm{8}=\mathrm{3}!+\mathrm{2}!=\sqrt{\left(\mathrm{3}!+\mathrm{2}!\right)^{\mathrm{2}} \:}\:=\:\sqrt{\mathrm{2}!\left(\mathrm{2}!+\mathrm{2}.\mathrm{3}!+\mathrm{3}.\mathrm{3}!\right)} \\ $$$$=\sqrt{\mathrm{2}!\left\{\mathrm{2}!+\mathrm{2}.\mathrm{3}!+\mathrm{3}!\left(\mathrm{4}−\mathrm{1}\right)\right\}}\:=\:\sqrt{\mathrm{2}!\left\{\mathrm{2}!+\mathrm{3}!+\mathrm{4}!\right)} \\ $$$$=\sqrt{\mathrm{2}!\left\{\mathrm{2}!+\sqrt{\left(\mathrm{3}!+\mathrm{4}!\right)^{\mathrm{2}} }\right.}=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!\left(\mathrm{3}!+\mathrm{2}.\mathrm{4}!+\mathrm{4}.\mathrm{4}!\right)}} \\ $$$$=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!\left\{\mathrm{3}!+\mathrm{2}.\mathrm{4}!+\mathrm{4}!\left(\mathrm{5}−\mathrm{1}\right)\right\}}}\: \\ $$$$=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!\left\{\mathrm{3}!+\left(\mathrm{4}!+\mathrm{5}!\right)\right.}}\: \\ $$$$=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!^{\mathrm{2}} +\mathrm{3}!\sqrt{\left(\mathrm{4}!+\mathrm{5}!\right)^{\mathrm{2}} }}}\:\:...... \\ $$$$=\sqrt{\mathrm{2}!^{\mathrm{2}} +\mathrm{2}!\sqrt{\mathrm{3}!^{\mathrm{2}} +\mathrm{3}!\sqrt{\mathrm{4}!^{\mathrm{2}} +\mathrm{4}!\sqrt{.......}}}} \\ $$$$\boldsymbol{\mathrm{Thanks}}\:\boldsymbol{\mathrm{Sir}}\:\boldsymbol{\mathrm{MJS}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{approximation}}\:\boldsymbol{\mathrm{help}}\:\boldsymbol{\mathrm{me}} \\ $$$$\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{lot}}.\:\boldsymbol{\mathrm{I}}\:\boldsymbol{\mathrm{appreciate}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{cooperation}}.\:\boldsymbol{\mathrm{Again}} \\ $$$$\boldsymbol{\mathrm{thanks}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{lot}}. \\ $$
Commented by MJS last updated on 22/Sep/19
$$\mathrm{great}!\:\mathrm{And}\:\mathrm{as}\:\mathrm{always},\:\mathrm{you}'\mathrm{re}\:\mathrm{welcome} \\ $$
Commented by ozodbek last updated on 23/Sep/19
$$\mathrm{thank}\:\mathrm{you} \\ $$
Commented by Prithwish sen last updated on 23/Sep/19
$$\mathrm{you}\:\mathrm{are}\:\mathrm{welcome}. \\ $$