If the geometric mea is `(1)/(n)` times the harmonic mean between two numbers, then show that the ratio of the two numbers is `1+sqrt(1-n^(2)):1-sqrt(1-n^(2))`.

Explanation

Let the two numbers be a and b.
Given, `G=(1)/(n)H " " "….(i)"`
Now, `G^(2)=AH`
`implies (H^(2))/(n^(2))=AH " " [ " From Eq. (i) " ]`
`therefore A=(H)/(n^(2))" " "…..(ii)"` Now, from important theorem of GM
`a,b=Apm sqrt((A^(2)-G^(2)))=(H)/(n^(2))pm sqrt(((H^(2))/(n^(4))-(H^(2))/(n^(2))))`
`(H)/(n^(2))[1pm sqrt((1-n^(2)))]`
`therefore (a)/(b)=((H)/(n^(2))[1+ sqrt((1-n^(2)))])/((H)/(n^(2))[1- sqrt((1-n^(2)))])`
`therefore a:b=1+sqrt((1-n^(2))):1-sqrt((1-n^(2)))`