Find the geometric means in each sequence 1 24 , _ , _ , _ , 54

Question

Find the geometric means in each sequence             1      24              ,      _              ,      _              ,      _        ,    54

Kristin Myers · Accepted Answer

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

a_{n} = a \cdot r^{n - 1}

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

a_{1} = \frac{1}{24}

and

a_{5} = 54

then we find:

r^{4} = \frac{a r^{4}}{a r^{0}} = \frac{a_{5}}{a_{1}} = \frac{54}{\frac{1}{24}} = 54 \cdot 24 = 1296 = 6^{4}

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

r = \pm 6

or

r = \pm 6 i

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:

\frac{1}{24}, \frac{1}{4}, \frac{3}{2}, 9, 54

Find the geometric means in each sequence 1/24,_,_,_,54

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