Let a,b,c be the numbers in GP.

Then we have the geometric mean = (a+c)^(0.5) = b

Then the terms become a, b+8, c which is in AP

Then arithmetic mean = (a+c)/2 = b+8.

Then the terms become a, b+8, c+64

Then geometric mean = (a+(c+64))^(0.5) = b+8

Now we have the solution as

a, (a + ((64a)^0.5/2)), (a + ((64a)^0.5/2))^(2)/a

Using trial and error and some common sense (Lol) for a=4 we get

a=4, b=12, c=36 which is in GP with 3 as the common ratio

Then

a=4, b=20, c=36 which is in AP with 16 as the common difference.

Then

a=4, b= 20, c=100 which is again in GP with 5 as the common ratio.

So a=4, b=12 & c=36 is a solution for the question.