Insert five geimetrec means between `(1)/(3)` and 9 and verify that their product is the fifth power of the geometric mean between `(1)/(3)` and 9.

Explanation

Let `G_(1),G_(2),G_(3),G_(4),G_(5)` be 5 GM's between `(1)/(3)` and 9.
Then, `(1)/(3),G_(1),G_(2),G_(3),G_(4),G_(5),9` are in GP.
Here, r= common ratio `=((9)/((1)/(3)))^((1)/(6))=3^((1)/(2))=sqrt(3)`
` therefore G_(1)=ar=(1)/(3)*sqrt(3) =(1)/(sqrt(3))`
` G_(2)=ar^(2)=(1)/(3)*3 =1`
` G_(3)=ar^(3)=(1)/(3)*3sqrt(3)=sqrt(3)`
` G_(4)=ar^(4)=(1)/(3)*9 =3`
` G_(5)=ar^(5)=(1)/(3)*9sqrt(3)=sqrt(3)`
Now, product `G_(1)xx G_(2)xxG_(3)xxG_(4)xxG_(5)`
`=(1)/(sqrt(3))xx1xxsqrt(3)xx3xx3sqrt(3)=9sqrt(3)=(3)^((5)/2)=(sqrt((1)/(3))xx9)^(5)`
`=[" GM of "(1)/(3) " and " 9^(5)]`