1. The problem is to find the smallest value (minimum) of the quadratic expression $$n^2 - 5n + 1$$. 2. A quadratic expression in the form $$an^2 + bn + c$$ with $$a > 0$$ opens upwards and its minimum value is at the vertex. 3. The vertex $$n$$ coordinate of a quadratic $$an^2 + bn + c$$ is given by $$n = -\frac{b}{2a}$$. 4. Here, $$a = 1$$ and $$b = -5$$, so the vertex is at: $$n = -\frac{-5}{2 \times 1} = \frac{5}{2} = 2.5$$. 5. To find the smallest value, substitute $$n = 2.5$$ back into the expression: $$\begin{aligned} (2.5)^2 - 5(2.5) + 1 &= 6.25 - 12.5 + 1 \\ &= -5.25 \end{aligned}$$ 6. Therefore, the smallest value of the expression $$n^2 - 5n + 1$$ is $$-5.25$$ at $$n = 2.5$$.