1. **State the problem:** We have a linear function $f$ with values given in a table: $$\begin{array}{c|c} x & f(x) \\\hline 1 & m \\ 2 & 6 \\ 3 & n \\ \end{array}$$ We need to find the value of $m + n$. 2. **Recall the property of linear functions:** A linear function has a constant rate of change (slope). This means the difference in $f(x)$ values divided by the difference in $x$ values is constant. 3. **Calculate the slope using known points:** Using points $(2,6)$ and $(1,m)$, $$\text{slope} = \frac{6 - m}{2 - 1} = 6 - m$$ Using points $(3,n)$ and $(2,6)$, $$\text{slope} = \frac{n - 6}{3 - 2} = n - 6$$ 4. **Set the slopes equal:** Since the slope is constant, $$6 - m = n - 6$$ 5. **Solve for $n$ in terms of $m$:** $$n = 12 - m$$ 6. **Find $m + n$:** $$m + n = m + (12 - m) = 12$$ **Final answer:** $$\boxed{12}$$