What are the geometric means between these terms?

First, I think you are supposed to find out all the 5 terms of the GS between 48 and 243/16

 

[243/16] / 48 ==0.31640625

 

0.31640625^(1/4) ==3 /4 - this is the common ratio.

 

The 5 terms of your GS look like this:

 

48 , 36 , 27 , 81 / 4 , 243 / 16 , >>Number of terms = 5>>Total Sum == 2343 / 16

 

Note: I don't think you need "geometric means"!!

I agree with the answers from guest; however, if you are to use geometric means:

 

The geometric mean of  a  and  b  is sqrt( a · b).

 

Therefore the geometric mean between  48  and  243/16  is:

     sqrt( 48 · 243/16 )  =  27      (the third term, or center term)

 

Then, the geometric mean between  48  and  27  is:

     sqrt( 48 · 27 )  =  36     (the second term)

 

and the geometric mean between  27  and  243/16  is:

    sqrt( 27 · 243/16 )  =  81/4     (the fourth term)