Question

How do you find the geometric means in each sequence `1/24, __, __, __, 54`?

Answer 1

The three geometric means are:

`1/4`, `3/2` and `9`

Explanation:

The general term of a geometric sequence can be written as:

`a_n = a*r^(n-1)`

where `a` is the initial term and `r` is the common ratio.

If `a_1 = 1/24` and `a_5 = 54` then we find:

`r^4 = (a r^4)/(a r^0) = a_5/a_1 = 54/(1/24) = 54*24 = 1296 = 6^4`

This has two Real solutions and two non-Real Complex solutions:

`r = +-6" "` or `" "r = +-6i`

Since the question asks about geometric means and the given terms are positive, we can probably assume that we want the positive common ratio `r = 6`.

Hence the sequence is:

`1/24, color(blue)(1/4), color(blue)(3/2), color(blue)(9), 54`