Question 1091492


if {{{a}}},{{{b}}},{{{c}}} are in {{{AP}}}, and {{{a}}},{{{mb}}},{{{c}}} are in {{{GP}}} then {{{a}}}, {{{m^2b}}}, {{{c}}} are in {{{HP }}}

{{{a}}},{{{b}}},{{{c}}} are in {{{AP}}} hence {{{2b = a+c}}}  -------------(1)
{{{a}}},{{{mb}}},{{{c}}} are in {{{GP}}} hence {{{m^2b^2 = ac}}}  ---------------(2)
divide (2) by (1)

{{{(m^2b^2)/2b = ac/(a+c)}}}

{{{(m^2b)/2 = ac/(a+c)}}}

{{{(m^2b)(a+c) = 2ac}}}

{{{(a+c)/ac = 2/(m^2b)}}}

{{{2/(m^2b)= 1/a + 1/c}}}

hence {{{a}}}, {{{m^2b}}}, {{{c}}} are in {{{HP }}}