Find four geometric means between 4096 and 972.

This means to make a geometric sequence with 6 terms with
the 1st term being 4096, and the 5th term 972, and four
terms (geometric means) between them. In other words,
you are to find the missing four numbers in

4096, ____, ____, ____, ____, 972

so that the six terms will be a geometric sequence.

We use the formula

an = a1rn-1

with n=6

a6 = a1r6-1

a6 = a1r5

Now we substitute a6 = 972, and a1 = 4096

972 = 4096r5

972/4096 = r5

243/1024 = r5

35/45 = r5

(3/4)5 = r5

Take fifth roots of both sides

3/4 = r

So we multiply the 1st term, a1, which is 4096,
by r, which is 3/4,  and get

a2 = 4096(3/4) = 3072

Then we multiply the 2nd term, a2, which is 3072,
by r, which is 3/4,  and get
  
a3 = 3072(3/4) = 2304

Then we multiply the 3rd term, a3, which is 2304,
by r, which is 3/4,  and get

a4 = 2304(3/4) = 1728

Then we multiply the 4th term, a4, which is 1728,
by r, which is 3/4,  and get

a5 = 1728(3/4) = 1296

Then finally, as a check, we multiply the 5th term, a5,
which is 1296, to see if we get the 6th term, which is
given as 972, by r, which is 3/4,  and see if we get
972.

a6 = 1296(3/4) = 972

Yes we do, so the geometric sequence is

4096, 3072, 2304, 1728, 1296, 972

and the four geometric means between the 1st and 6th terms are
the four terms between them:

3072, 2304, 1728, and 1296 

Edwin
AnlytcPhil@aol.com