Answer :
ANSWER
12, 36, 108 and 324.
EXPLANATION
We want to find 4 geometric means between 4 and 972.
Let w, x, y and z be the four geometric means.
So, we have that:
4, w, x, y, z, 972
The first term, a, of this progression is 4.
The 6th term is 972
We have that the nth term of a geometric progression is:
[tex]a_n=ar^{n\text{ - 1}}[/tex]Let us find r by using the 6th term:
[tex]\begin{gathered} 972\text{ = 4 }\cdot r^{6\text{ - 1}}\text{ = 4 }\cdot r^5 \\ r^5\text{ = }\frac{972}{4}\text{ = 243} \\ r\text{ = }\sqrt[5]{243} \\ r\text{ = }3 \end{gathered}[/tex]Now, let us find the four terms:
=> w is the second term, so n = 2:
[tex]\begin{gathered} w\text{ = 4 }\cdot3^{2\text{ - 1}}\text{ = 4 }\cdot\text{ 3} \\ w\text{ = 12} \end{gathered}[/tex]=> x is the thrid term, so n = 3:
[tex]\begin{gathered} x\text{ = 4 }\cdot3^{3\text{ - 1}}\text{ = 4 }\cdot3^2\text{ = 4 }\cdot\text{ 9 } \\ x\text{ = 36} \end{gathered}[/tex]=> y is the fourth term, so, n = 4:
[tex]\begin{gathered} y\text{ = 4 }\cdot3^{4\text{ - 1}}\text{ = 4 }\cdot3^3\text{ = 4 }\cdot\text{ 27} \\ y\text{ = 108} \end{gathered}[/tex]=> z is the fifth term, so n =5:
[tex]\begin{gathered} z\text{ = 4 }\cdot3^{5\text{ - 1}}\text{ = 4 }\cdot3^4\text{ = 4 }\cdot\text{ 81} \\ z\text{ = 324} \end{gathered}[/tex]So, the four geometric means are 12, 36, 108 and 324.
