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.