Question

Find the four geometric means between 4 and 972. 4,__,__,__,__,972.

Answer 1

The four geometric means between 4 and 972 are 12, 36, 108, and 324.

Step-by-step explanation:

To find the four geometric means between 4 and 972, we need to find the common ratio. The common ratio (r) is found by taking the n-th root of the final term divided by the first term. In this case, the n-th root is the fourth root because we want to find four geometric means. So, r = ∛(972/4) = 3.

To find the geometric means, we multiply the first term by the common ratio raised to the power of the term number. The four geometric means are:

4, 12, 36, 108, 324, 972

Answer 2

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:

`a_n=ar^{n\text{ - 1}}`

Let us find r by using the 6th term:

`\begin{gathered} 972\text{ = 4 }\cdot r^{6\text{ - 1}}\text{ = 4 }\cdot r^5 \\ r^5\text{ = }(972)/(4)\text{ = 243} \\ r\text{ = }\sqrt[5]{243} \\ r\text{ = }3 \end{gathered}`

Now, let us find the four terms:

=> w is the second term, so n = 2:

`\begin{gathered} w\text{ = 4 }\cdot3^{2\text{ - 1}}\text{ = 4 }\cdot\text{ 3} \\ w\text{ = 12} \end{gathered}`

=> x is the thrid term, so n = 3:

`\begin{gathered} x\text{ = 4 }\cdot3^{3\text{ - 1}}\text{ = 4 }\cdot3^2\text{ = 4 }\cdot\text{ 9 } \\ x\text{ = 36} \end{gathered}`

=> y is the fourth term, so, n = 4:

`\begin{gathered} y\text{ = 4 }\cdot3^{4\text{ - 1}}\text{ = 4 }\cdot3^3\text{ = 4 }\cdot\text{ 27} \\ y\text{ = 108} \end{gathered}`

=> z is the fifth term, so n =5:

`\begin{gathered} z\text{ = 4 }\cdot3^{5\text{ - 1}}\text{ = 4 }\cdot3^4\text{ = 4 }\cdot\text{ 81} \\ z\text{ = 324} \end{gathered}`

So, the four geometric means are 12, 36, 108 and 324.