geometric sequence

 

We have two geometric sequences of positive real numbers:     

6,a,b and a,b,54     

Solve for a      

 

6               =  1st  term     

6 • r           =  2^nd  term ... this is a     

6 • r • r      =  3^rd  term ... this is b     

6 • r • r • r  =  4^th  term ... this is 54     

 

so                 6r³  =  54    

                       r³  =  9     

                       r    =  cube root of 9     

 

a  =  6 • r  =  6 • cbrt(9)     

 

This is about 12.48 but that's just a close approximation.   

_.