Marquis begins a job making \( \$ 53,000 \) per year. Each year, he is to receive a \( \$ 1,600 \) raise. His salary after \( n \) years can be modeled by this equation: \( a_{n}=53,000+(n-1) 1,800 \). After 25 years, how much total money will Marquis have earned from this job? \( \$ 1,792,000 \) \( \$ 1,805,000 \) \( \$ 1,844,975 \) \( \$ 1,845,000 \)

To find out how much total money Marquis will have earned after 25 years, we first need to calculate his salary for each year for 25 years using the provided formula \( a_{n} = 53,000 + (n - 1) \times 1,600 \). For \( n = 25 \): \[ a_{25} = 53,000 + (25 - 1) \times 1,600 \] \[ a_{25} = 53,000 + 24 \times 1,600 = 53,000 + 38,400 = 91,400 \] Now, we can find the total earnings over these 25 years. To do this, we can sum up all the individual yearly salaries from year 1 to year 25. The total money earned can be calculated as: \[ \text{Total Earnings} = a_1 + a_2 + a_3 + \ldots + a_{25} \] Instead of adding each year separately, we can use the formula for the sum of an arithmetic series, which is: \[ \text{Total Earnings} = \frac{n}{2} \times (a_1 + a_{n}) \] where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term. Here: - \( n = 25 \) - \( a_1 = 53,000 \) - \( a_{25} = 91,400 \) Calculating the total earnings: \[ \text{Total Earnings} = \frac{25}{2} \times (53,000 + 91,400) = 12.5 \times 144,400 = 1,805,000 \] Therefore, after 25 years, Marquis will have earned a total of **$1,805,000**.