If n geometric means between a and b be #G_1,G_2,G_3,.......,G_n,# and a geometric mean be G Then the value of #G_1*G_2*G_3*........*G_n#=(in terms of G)?

1 Answer
Sep 22, 2017

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#