If a be one A.M and `G_1` and `G_2` be then geometric means between b and c then `G_1^3+G_2^3=`

Explanation

Given, a=AM between b and c
`implies " " a=(b+c)/(2) implies 2a=b+c " " "………(i)"`
Again, `b,G_(1),G_(2),c` are in GP.
` therefore " " (G_(1))/(b)=(G_(2))/(G_(1))=(c)/(G_(2)) implies b=(G_(1)^(2))/(G_(2)),c=(G_(2)^(2))/(G_(1))`
and `G_(1)G_(2)=bc " " ".......(ii)"`
From Eqs. (i) and (ii),
`2a= (G_(1)^(2))/(G_(1))+(G_(2)^(2))/(G_(2))=(G_(1)^(3)+ G_(2)^(3))/(G_(1)G_(2))=(G_(1)^(3)+G_(2)^(3))/(bc)[therefore G_(1)G_(2)=bc]`
`implies G_(1)^(3)+G_(2)^(3)=2abc`