Alg8.14

Insert $5\,$ geometric means between $\frac{1}{3}\,$ and $243\,$

Let the five geometric means be $x_1,x_2,x_3,x_4,x_5\,$

Therefore $\frac{1}{3},x_1,x_2,x_3,x_4,x_5,243\,$ are in GP.

Total number of terms is $7\,$

Therefore $a=\frac{1}{3},t_7=ar^6=243,\frac{1}{3}r^6=243,r^6=243\cdot 3=729=3^6,r=3\,$

Hence $x_1=ar=\frac{1}{3}\cdot 3=1\,$

$x_2=ar^2=\frac{1}{3}\cdot 3^2=3\,$

$x_3=ar^3=\frac{1}{3}\cdot 3^3=9\,$

$x_4=ar^4=\frac{1}{3}\cdot 3^4=27\,$

$x_5=ar^5=\frac{1}{3}\cdot 3^5=81\,$

Therefore,the geometric means between the given two numbers is $1,3,9,27,81\,$ respectively.