To find 3 geometric means between 256 and 1, you can use the formula for the geometric sequence:

$a_n = a_1 \cdot r^{n-1}$

Where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term (256),
• $r$ is the common ratio,
• $n$ is the position of the term.

We need to find 3 geometric means, which implies that there are 5 terms in total: $a_1 = 256$, then 3 geometric means, and $a_5 = 1$.

The formula for the last term of a geometric sequence can be written as:

$a_5 = a_1 \cdot r^4$

Substituting the known values:

$1 = 256 \cdot r^4$

Solve for $r^4$:

$r^4 = \frac{1}{256}$

$r^4 = 256^{-1} = (2^8)^{-1} = 2^{-8}$

$r = \left(2^{-8}\right)^{\frac{1}{4}} = 2^{-2} = \frac{1}{4}$

So, the common ratio $r = \frac{1}{4}$.

Now, we can find the geometric means. The sequence is:

$a_1 = 256, \quad a_2 = 256 \times \frac{1}{4}, \quad a_3 = a_2 \times \frac{1}{4}, \quad a_4 = a_3 \times \frac{1}{4}, \quad a_5 = 1$

Let's calculate these terms:

1. $a_2 = 256 \times \frac{1}{4} = 64$,
2. $a_3 = 64 \times \frac{1}{4} = 16$,
3. $a_4 = 16 \times \frac{1}{4} = 4$.

Thus, the 3 geometric means between 256 and 1 are 64, 16, and 4.

Would you like further details on this method or have any questions?

Here are some related questions to deepen your understanding:

1. What is the general formula for finding geometric means between two numbers?
2. How do you determine the common ratio in a geometric sequence?
3. How does a geometric sequence differ from an arithmetic sequence?
4. What happens if you change the number of geometric means you want to find between two numbers?
5. Can you apply this method to find geometric means between any two numbers?

Tip: The geometric mean of two numbers $a$ and $b$ can also be found as $\sqrt{a \cdot b}$.