Question: Micah takes a job with a starting salary of $78,000 for the first year. He earns a 4% increase each…

Micah takes a job with a starting salary of $78,000 for the first year. He earns a 4% increase each year. To the nearest dollar, how much does Micah make over the first seven years?

$616,067 $567,840 $718,710 $718,499

Solution

Micah starts with a salary of $78,000 and receives a 4% increase each year. We need to find the total amount earned over the first seven years. 1. The salary for each year can be calculated using the formula for geometric progression: \[ \text{Salary for year } n = \text{Initial Salary} \times (1 + \text{rate of increase})^{n-1} \] 2. Calculate the salary for each of the seven years and sum them up: \[ \begin{align*} \text{Year 1:} & \quad 78000 \times (1 + 0.04)^0 = 78000 \\ \text{Year 2:} & \quad 78000 \times (1 + 0.04)^1 = 81120 \\ \text{Year 3:} & \quad 78000 \times (1 + 0.04)^2 = 84364.8 \\ \text{Year 4:} & \quad 78000 \times (1 + 0.04)^3 = 87739.392 \\ \text{Year 5:} & \quad 78000 \times (1 + 0.04)^4 = 91248.96768 \\ \text{Year 6:} & \quad 78000 \times (1 + 0.04)^5 = 94903.9263872 \\ \text{Year 7:} & \quad 78000 \times (1 + 0.04)^6 = 98600.0834427 \\ \end{align*} \] 3. Sum the salaries over seven years: \[ \text{Total} = 78000 + 81120 + 84364.8 + 87739.392 + 91248.96768 + 94903.9263872 + 98600.0834427 \] 4. Calculate the sum: \[ \text{Total} = 616,067 \] Therefore, Micah earns approximately $616,067 over the first seven years.