We are asked to order the following numbers from least to greatest: $\frac{9}{2}$, $4.\overline{23}$, and $\sqrt{17}$.

1. Problem Description

We are asked to order the following numbers from least to greatest: $\frac{9}{2}$, $4.\overline{23}$, and $\sqrt{17}$.

2. Solution Steps

First, let's convert each number into a decimal representation to compare them easily.

$\frac{9}{2} = 4.5$

$4.\overline{23} = 4.232323...$

We know that $4^2 = 16$ and $5^2 = 25$, so $\sqrt{17}$ will be between 4 and

5. We can estimate it using a calculator or by knowing that $4.1^2 = 16.81$ and $4.2^2 = 17.64$. Therefore, $\sqrt{17}$ is between 4.1 and 4.

2. We can further refine it using a calculator, $\sqrt{17} \approx 4.123105626...$

Now we compare the decimal representations:

$4.232323... < 4.5$

$4.123105626... < 4.232323... < 4.5$

So the order from least to greatest is $\sqrt{17}$, $4.\overline{23}$, $\frac{9}{2}$.

3. Final Answer

$\sqrt{17}, 4.\overline{23}, \frac{9}{2}$