Insert 4 geometric means between 72 and 9/4

To insert 4 geometric means between two numbers, \( a_1 = 72 \) and \( a_6 = \frac{9}{4} \), we first need to understand that these numbers will form a geometric sequence. The general form of a geometric sequence is:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Where:
- \( a_n \) is the \( n \)-th term of the sequence.
- \( r \) is the common ratio.
- \( n \) is the position of the term in the sequence.

Given \( a_1 = 72 \) and \( a_6 = \frac{9}{4} \), we have:

\[
a_6 = a_1 \cdot r^{(6-1)} = 72 \cdot r^5 = \frac{9}{4}
\]

We need to solve for \( r \):

\[
r^5 = \frac{\frac{9}{4}}{72} = \frac{9}{4} \times \frac{1}{72} = \frac{9}{288} = \frac{1}{32}
\]

Taking the 5th root of both sides:

\[
r = \left(\frac{1}{32}\right)^{\frac{1}{5}}
\]

Since \( 32 = 2^5 \), we have:

\[
r = \frac{1}{2}
\]

Now that we have the common ratio \( r = \frac{1}{2} \), we can find the four geometric means between 72 and \( \frac{9}{4} \):

1. \( a_2 = a_1 \cdot r = 72 \times \frac{1}{2} = 36 \)
2. \( a_3 = a_2 \cdot r = 36 \times \frac{1}{2} = 18 \)
3. \( a_4 = a_3 \cdot r = 18 \times \frac{1}{2} = 9 \)
4. \( a_5 = a_4 \cdot r = 9 \times \frac{1}{2} = 4.5 \)

Thus, the geometric sequence is \( 72, 36, 18, 9, 4.5, \frac{9}{4} \).

### Summary
The 4 geometric means between 72 and \( \frac{9}{4} \) are: \( 36, 18, 9, \) and \( 4.5 \).

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### Related Questions
1. How do you calculate the geometric mean of two numbers?
2. What is the difference between arithmetic and geometric sequences?
3. Can a geometric sequence contain negative numbers?
4. How does the geometric mean relate to exponential growth?
5. How would you insert geometric means between negative numbers?

### Tip
When solving for a common ratio in a geometric sequence, always check if the result is a rational or an irrational number, as this impacts the type of sequence formed.

Learn how to insert 4 geometric means between the numbers 72 and 9/4 using the general form of a geometric sequence. This problem explores geometric sequences and includes detailed steps to find the common ratio and calculate each mean. Suitable for high school students studying sequences and series.

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