1. Problem a: Find total future profits of a tool with a net return decreasing by 10% annually and a current gain of 400. 2. Since net return decreases by 10% annually, the common ratio $r=1-0.10=0.9$. 3. The net gain for the current year (first term of G.P.) is $a=400$. 4. Total future profits (infinite sum of a G.P.) is $S_{\infty}=\frac{a}{1-r}=\frac{400}{1-0.9}=\frac{400}{0.1}=4000$. --- 5. Problem b: Find 6 speeds in G.P. starting at 50 rev/min and ending at 750 rev/min. 6. Let the speeds be $a, ar, ar^2, ar^3, ar^4, ar^5$, where $a=50$ and $ar^5=750$. 7. Solve for $r$: $$ar^5=750 \Rightarrow 50r^5=750 \Rightarrow r^5=\frac{750}{50}=15 \Rightarrow r=15^{\frac{1}{5}}.$$ 8. Calculate $r$: $$r\approx 15^{0.2} \approx 1.718.$$ 9. Calculate speeds to nearest whole number: - $a=50$ - $50 \times 1.718 \approx 86$ - $50 \times 1.718^2 \approx 148$ - $50 \times 1.718^3 \approx 254$ - $50 \times 1.718^4 \approx 435$ - $50 \times 1.718^5 = 750$ (given) Final speeds: 50, 86, 148, 254, 435, 750 rev/min.