Resolve the problem. Justify your answer, using the formulas of both arithmetic progression and geometric progression. Jaime and Rafael are dedicated to painting houses. Jaime quotes $24 for the first hour and then adds $2.85 for each hour of work. In this way, the second hour is charged at $26.85 and so on. On the other hand, Rafael quotes $1.50 for the first hour and then doubles the cost per hour. After a 7-hour work day. What type of progression does each painter's hourly rate represent? Who would earn more, adding a total of 7 hours of work?

*1. Identify the type of progression for each painter:* Jaime: - First hour: $24 - Second hour: $24 + $2.85 = $26.85 - Third hour: $26.85 + $2.85 = $29.70 ... and so on. The difference between consecutive terms is constant, which is $2.85. Thus, Jaime's hourly rate represents an *arithmetic progression (AP)*. Rafael: - First hour: $1.50 - Second hour: $1.50 × 2 = $3 - Third hour: $3 × 2 = $6 ... and so on. The ratio between consecutive terms is constant, which is 2. Thus, Rafael's hourly rate represents a *geometric progression (GP)*. *2. Calculate the total earnings for each painter for 7 hours using the formulas for AP and GP:* For Jaime (AP): Sn = (n/2) (2a + (n-1)d) Where: Sn is the sum of the first n terms n is the number of terms (7 in this case) a is the first term $24 d is the common difference $2.85 \[ S_7 = \frac{7}{2} (2(24) + (7-1)(2.85)) \] \[ S_7 = \frac{7}{2} (48 + 17.1) \] S7 = 7/2 × 65.1 = 227.85 For Rafael (GP): Sn = a ((r^n) - 1)/(r - 1) Where: Sn) is the sum of the first n terms n is the number of terms (7 in this case) a is the first term ($1.50) r is the common ratio (2) S7 = 1.5 ((2^7) - 1)/(2 - 1) S7 = 1.5 × (127) = 190.5 *3. Compare the total earnings:* Jaime: $227.85 Rafael: $190.5 *Conclusion:* - Jaime's hourly rate represents an arithmetic progression. - Rafael's hourly rate represents a geometric progression. - After a 7-hour workday, Jaime would earn more with a total of $227.85 compared to Rafael's $190.5.