Let $${{{b}}_{{i}}} > 1{\text{ for }} i = 1,2,....,101.$$     Suppose $${\log _e}{{{b}}_1},{\log _e}{{{b}}_2},.....,{\log _e}{{{b}}_{101}}$$      are in Arithmetic Progression (A.P.) with the common difference $${\log _e}2$$. Suppose $${{{a}}_1},{{{a}}_2},.....{{,}}{{{a}}_{101}}$$    are in A.P. such that $${{{a}}_1} = {{{b}}_1}$$   and $${{{a}}_{51}} = {{{b}}_{51}}$$ . If $${{t}} = {{{b}}_1} + {{{b}}_2} + ..... + {{{b}}_{51}}$$     and $${{s}} = {{{a}}_1} + {{{a}}_2} + ..... + {{{a}}_{51}},$$     then

Solution :
$$\eqalign{ & {\log _{{e}}}{{{b}}_1},{\log _{{e}}}{{{b}}_2}, - - - ,{\log _{{e}}}{{{b}}_{{{101}}}}\,{\text{are}}\,{\text{in}}\,{\text{A}}.{\text{P}}. \cr & \Rightarrow \,\,{b_1},{{{b}}_2}, - - - {{{b}}_{{{101}}}}\,{\text{are}}\,{\text{in}}\,{\text{G}}{\text{.P}}{\text{.}} \cr & {\text{Also}}\,{{{a}}_1},\,{{{a}}_2}, - - - {{{a}}_{{{101}}}}\,{\text{are}}\,{\text{in}}\,{\text{A}}{\text{.P}}{\text{.}} \cr & {\text{where}}\,{{{a}}_1} = {{{b}}_{{1}}}\,{\text{are}}\,{{{a}}_{{{51}}}}{{ = }}{{{b}}_{{{51}}}}{{.}} \cr & \therefore \,\,\,{{{b}}_2},{{{b}}_3}, - - - ,{{{b}}_{50}}{\text{ and }}{\text{GM's}}\,\,{{{a}}_2},{{{a}}_3}, - - - ,{{{a}}_{50}}{\text{ are in AM's}} \cr & {\text{between }}{{{b}}_1}{\text{ and }}{{{b}}_{51}}. \cr & \because {\text{ GM}} < {\text{AM}} \cr & \Rightarrow \,\,{{{b}}_2} < {{{a}}_2},{{{b}}_3} < {{{a}}_3}, - - - ,{{{b}}_{50}} < {{{a}}_{50}} \cr & \because \,\,\,\,{{{b}}_1} + {{{b}}_2} + - - - + {{{b}}_{51}} < {{{a}}_1} + {{{a}}_2} + - - - + {{{a}}_{51}} \cr & \Rightarrow \,\,\,{{t}}\, < {{s}} \cr & {\text{Also }}{{{a}}_1},{{{a}}_{51}},{{{a}}_{101}}{\text{ are in A}}{\text{.P}}{\text{.}} \cr & \,\,\,\,\,\,\,\,\,\,\,{{{b}}_1},{{{b}}_{51}},{{{b}}_{101}}{\text{ are in G}}{\text{.P}}{\text{.}} \cr & \because \,\,\,\,\,\,\,{{{a}}_1} = {{{b}}_1}{\text{ and }}{{{a}}_{51}} = {{{b}}_{51}} \cr & \therefore \,\,\,\,\,\,\,{{{b}}_{101}} > {{{a}}_{101}} \cr} $$