If `G_1` and `G_2` are two geometric means and A is the arithmetic mean inserted two numbers, then the value of `(G_1^2)/G_2+(G_2^2)/G_1` is:

Explanation

Let the numbers be a and b .
then , `A=(a+b)/(2)`
`implies 2A=a+b`
and` G_(1),G_(2)` be geometric mean between a and b then a `G_(1).g_(2)` b are in GP.
let r be the common ratio
then `b=ar^(4-1) [:' a_(n)=ar^(n-1)]`
`implies b=ar^(3) implies (b)/(a)=r^(3)`
`therefore r=((b)/(a))^(1//3)`
Now ` G_(1) =ar =a((b)/(a))^(1//3) [ :' r=((b)/(a))^(1//3)]`
`and G_(2) =ar^(2)=a((b)/(a))^(2//3)`
`RHS =(G_(1)^(2))/(G_(2))=([a((b)/(a))^(1//3)]^(2))/(a((b)/a)^(2//3))+([a((b)/(a))^(2//3)]^(2))/(a((b)/(a))^(1//3))`
`=(a^(2)((b)/(a))^(2//3))/(a((b)/(a))^(2//3))+(a^(2)((b)/(a))^(4//3))/(a((b)/(a))^(1//3))`
` =a+a((b)/(a))=a+b=2A ["using Eq (i) ]`
`=LHS`