To find two geometric means between 7 and 189, we can follow the process of inserting two numbers in a geometric sequence. Let the four terms of the sequence be:

\[ 7, G_1, G_2, 189 \]

Since the numbers form a geometric sequence, the ratio between consecutive terms must be constant. Let that ratio be \( r \).

Thus, we have the following relations:
\[
G_1 = 7r, \quad G_2 = 7r^2, \quad 189 = 7r^3
\]

Now, solve for \( r \):
\[
189 = 7r^3 \implies r^3 = \frac{189}{7} = 27 \implies r = \sqrt[3]{27} = 3
\]

Now that we know \( r = 3 \), we can find the two geometric means:

\[
G_1 = 7r = 7 \times 3 = 21
\]
\[
G_2 = 7r^2 = 7 \times 3^2 = 7 \times 9 = 63
\]

Thus, the two geometric means between 7 and 189 are **21** and **63**.

Would you like further details or have any questions?

Here are five related questions you might find interesting:
1. How do you find three geometric means between two numbers?
2. What is the formula for the \(n\)-th term of a geometric sequence?
3. How do arithmetic means differ from geometric means?
4. How do you determine the common ratio if three terms are given in a geometric sequence?
5. Can geometric means be negative?

**Tip**: In a geometric sequence, the ratio between consecutive terms is constant, and multiplying the first term by the common ratio raised to different powers gives the terms.

Find 2 geometric means between 7 and 189.

Learn how to find two geometric means between 7 and 189 using the geometric sequence formula. This step-by-step solution explains the process of calculating the common ratio and applying it to find the missing terms. Suitable for students in Grades 9-10.

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