Inserting #n# geometric means between #a and b # we get the following GP series of total #(n+2)# terms
#a,G_1,G_2,G_3,..........G_n,b#
Let #r# be the common ratio of this GP then #b# becomes the #n+2# th term of the series. So we have
#b=ar^(n+1)#
#=>r=(b/a)^(1/(n+1))........[1]#
Again G is the single GM between #a and b #, So we have
#G=(ab)^(1/2)#
#=>ab=G^2.......[2]#
And the product
#G_1xxG_2xxG_3xx..........xxG_n#
#=prod_(i=1)^(i=n)G_i=prod_(i=1)^(i=n)ar^i=a^nr^(sum_(i=1)^(i=n)i=a^nr^((n(n+1))/2)#
#=a^nxx((b/a)^(1/(n+1)))^((n(n+1))/2)# #" "color(red)("Inserting "r=(b/a)^(1/(n+1)))#
#=a^nxx(b/a)^(n/2)#
#=a^(n/2)xxb^(n/2)#
#=(ab)^(n/2)#
#=(G^2)^(n/2)# #" "color(red)("Inserting " ab=G^2#
#=G^n#
