1. **Problem (a):** A hire tool firm's net return decreases by 10% annually. The current net gain is 400. Find the total of all future profits assuming the tool lasts forever. 2. Since the net return decreases by 10% per year, the common ratio of the geometric series is $r=1-0.10=0.90$. 3. The first term $a=400$ represents the net gain this year. 4. The total of all future profits is the sum of an infinite geometric series: $$ S=\frac{a}{1-r} = \frac{400}{1-0.90} = \frac{400}{0.10} = 4000 $$ 5. **Problem (b):** A drilling machine has 6 speeds from 50 rev/min to 750 rev/min forming a geometric progression. Find all speeds. 6. Let the first term be $a=50$ and last term be $l=750$, number of terms $n=6$, and common ratio $r$. 7. The $n$th term of a geometric sequence is given by: $$ l = a r^{n-1} $$ Substitute values: $$ 750 = 50 r^{5} $$ 8. Solve for $r$: $$ r^{5} = \frac{750}{50} = 15 $$ $$ r = 15^{\frac{1}{5}} $$ 9. Compute $r$: $$ r \approx 15^{0.2} \approx 1.717 \text{ (rounded to 3 decimal places)} $$ 10. Calculate all speeds: - $a_1 = 50$ - $a_2 = 50 \times 1.717 \approx 86$ - $a_3 = 86 \times 1.717 \approx 148$ - $a_4 = 148 \times 1.717 \approx 254$ - $a_5 = 254 \times 1.717 \approx 436$ - $a_6 = 436 \times 1.717 \approx 750$ 11. Round speeds to nearest whole number: 50, 86, 148, 254, 436, 750. **Final answers:** - (a) Total future profits: $4000$ - (b) Speeds: $50, 86, 148, 254, 436, 750$