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

1 Answer
Jan 1, 2017

Answer:

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#