Find the two geometric means between 2 and 16

To find the two geometric means between 2 and 16, we can use the formula for the geometric mean in a sequence. If we have a sequence \(a_1, a_2, a_3, a_4\) where \(a_1 = 2\), \(a_4 = 16\), and \(a_2\) and \(a_3\) are the geometric means, the relationship between consecutive terms in a geometric sequence is given by:

\[
\text{Common ratio} = r = \frac{a_{n+1}}{a_n}
\]

Since there are two geometric means, the sequence can be written as:

\[
2, a_2, a_3, 16
\]

Let the common ratio be \(r\). Then:

\[
a_2 = 2r
\]
\[
a_3 = 2r^2
\]
\[
16 = 2r^3
\]

Now, solve for \(r\):

\[
r^3 = \frac{16}{2} = 8
\]

\[
r = \sqrt[3]{8} = 2
\]

Now, we can find the geometric means:

\[
a_2 = 2r = 2 \times 2 = 4
\]

\[
a_3 = 2r^2 = 2 \times 4 = 8
\]

So, the two geometric means between 2 and 16 are **4** and **8**.

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. What is the geometric mean of two numbers?
2. How do you find the nth term in a geometric sequence?
3. Can geometric means be negative?
4. How do you solve for the common ratio in a geometric sequence?
5. What is the difference between arithmetic and geometric sequences?

**Tip:** The geometric mean is always greater than or equal to the arithmetic mean if all the numbers in the sequence are positive.

Learn how to find the two geometric means between 2 and 16 using geometric sequences and ratios. This problem involves calculating the common ratio and applying it to solve for the geometric means. Suitable for students in Grades 9-11.

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