To solve for $r$ in the equation

$n = \frac{1 + r^2}{2},$

follow these steps:

1. Multiply both sides by 2 to eliminate the fraction:

$2n = 1 + r^2.$

2. Subtract 1 from both sides:

$2n - 1 = r^2.$

3. Finally, take the square root of both sides to solve for $r$:

$r = \sqrt{2n - 1} \quad \text{or} \quad r = -\sqrt{2n - 1}.$

Thus, the solutions for $r$ are:

$r = \pm \sqrt{2n - 1}.$

If you have any further questions or need more details, feel free to ask! Here are some related questions:

1. What are the conditions for $n$ for $r$ to be a real number?
2. How does changing $n$ affect the value of $r$?
3. Can you provide examples with specific values of $n$?
4. How would you graph the relationship between $n$ and $r$?
5. What other mathematical concepts relate to this equation?

Tip: When solving for variables in equations, always check the domain of the solution to ensure that it meets any necessary conditions.