Identify common factors in the numerator.

The numerator is $5^n - 5^{n-2}$. We can factor out $5^{n-2}$.
Recall that $5^n = 5^{n-2} \cdot 5^2$.

$$5^n - 5^{n-2} = 5^{n-2} \cdot 5^2 - 5^{n-2} \cdot 1 = 5^{n-2}(5^2 - 1)$$ $$5^{n-2}(25 - 1) = 5^{n-2}(24)$$

The denominator is $5^3 \times 5^n - 125 \times 5^{n-2}$.
First, calculate $5^3$: $5^3 = 5 \times 5 \times 5 = 125$.
Substitute this value into the denominator:

$$125 \times 5^n - 125 \times 5^{n-2}$$

Now, factor out 125:

$$125(5^n - 5^{n-2})$$

From Step 1, we know that $5^n - 5^{n-2} = 5^{n-2}(24)$. Substitute this into the denominator:

$$125 \cdot 5^{n-2}(24)$$

Both the numerator and denominator have the term $5^{n-2}(24)$.

$$\frac{\cancel{5^{n-2}(24)}}{125 \cdot \cancel{5^{n-2}(24)}} = \frac{1}{125}$$

The simplified expression is:

$$\frac{1}{125}$$