1. **Problem:** The mean of the numbers $m$, $8m+1$, $17$, and $20$ is $14$. Find (a) the value of $m$ and (b) the mode. 2. **Formula for mean:** The mean of $n$ numbers $x_1, x_2, ..., x_n$ is given by $$\text{Mean} = \frac{x_1 + x_2 + \cdots + x_n}{n}$$ 3. **Step (a) Find $m$:** The numbers are $m$, $8m+1$, $17$, and $20$. Their mean is given as $14$, so $$\frac{m + (8m+1) + 17 + 20}{4} = 14$$ 4. Simplify numerator: $$m + 8m + 1 + 17 + 20 = 9m + 38$$ 5. Substitute back: $$\frac{9m + 38}{4} = 14$$ 6. Multiply both sides by 4: $$\cancel{4} \times \frac{9m + 38}{\cancel{4}} = 14 \times 4$$ $$9m + 38 = 56$$ 7. Subtract 38 from both sides: $$9m + 38 - 38 = 56 - 38$$ $$9m = 18$$ 8. Divide both sides by 9: $$\frac{9m}{\cancel{9}} = \frac{18}{\cancel{9}}$$ $$m = 2$$ 9. **Step (b) Find the mode:** The numbers are $m=2$, $8m+1=8(2)+1=17$, $17$, and $20$. The numbers are $2$, $17$, $17$, and $20$. 10. The mode is the number that appears most frequently. Here, $17$ appears twice, others once. **Mode = 17** **Final answers:** (a) $m = 2$ (b) Mode = $17$