To solve this problem, we will first find the value of $x$ using the arithmetic mean and then use it to calculate the geometric mean.

Step 1: Find $x$

The arithmetic mean between 4 and $x$ is given by:

$\text{Arithmetic Mean} = \frac{4 + x}{2}$

We are told that this arithmetic mean is 34:

$\frac{4 + x}{2} = 34$

Now, solve for $x$:

$4 + x = 68$

$x = 68 - 4$

$x = 64$

Step 2: Calculate the Geometric Mean

The geometric mean of 4 and $x$ (which is now 64) is given by:

$\text{Geometric Mean} = \sqrt{4 \times 64}$

$\text{Geometric Mean} = \sqrt{256}$

$\text{Geometric Mean} = 16$

So, the geometric mean is $16$.

Would you like to explore the relationship between arithmetic and geometric means further? Here are some related questions:

1. What are the conditions under which the arithmetic mean equals the geometric mean?
2. How do arithmetic and geometric means compare when all terms in the sequence are positive?
3. Can the geometric mean be negative, and if so, under what conditions?
4. What is the significance of the geometric mean in finance or data analysis?
5. How does changing one of the numbers in the sequence affect both the arithmetic and geometric means?

Tip: The arithmetic mean is always greater than or equal to the geometric mean for any set of non-negative numbers, a property known as the AM-GM inequality.