Three numbers are in geometric progression. Their sum is 26 and their product is 216. Find the numbers.

Let the three terms of the geometric progression be $\frac{a}{r}$, $a$, and $ar$, where $a$ is the middle term and $r$ is the common ratio.
The sum of the first three terms is 26:

The product of the first three terms is 216:

Take the cube root of both sides:

Substitute $a=6$ into equation $(*)$:

Subtract 6 from both sides:

Multiply the entire equation by $r$ to eliminate the fraction:

Rearrange into a standard quadratic equation form $Ar^2 + Br + C = 0$:

Divide the entire equation by 2 to simplify:

Factor the quadratic equation:

Set each factor to zero to find the possible values for $r$:

The common ratio can be $\frac{1}{3}$ or 3.

The common ratio is $\boxed{\frac{1}{3} \text{ or } 3}$.

The difference of squares formula states that $x^2 - y^2 = (x - y)(x + y)$.

We are given:

Substitute these values into the formula:

Divide both sides by 2:

The value of $x + y$ is $\boxed{6}$.