Question: Marquis begins a job making $53,000 per year. Each year, he is to receive a $1,600 raise. His…

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: \( a_n = 53,000 + (n - 1)1,600 \). After 25 years, how much total money will Marquis have earned from this job?

$1,805,000

$1,845,000

$1,792,000

$1,844,975

Solution

Marquis’s salary after \( n \) years is given by the equation: \[ a_n = 53,000 + (n - 1) \times 1,600 \] To find out how much total money Marquis will have earned after 25 years, we need to calculate the sum of his salary from year 1 to year 25. First, calculate the salary for year 1: \[ a_1 = 53,000 \] Next, calculate the salary for year 25: \[ a_{25} = 53,000 + (25 - 1) \times 1,600 \] \[ a_{25} = 53,000 + 24 \times 1,600 \] \[ a_{25} = 53,000 + 38,400 \] \[ a_{25} = 91,400 \] The total money earned is the sum of an arithmetic series from year 1 to year 25. The sum \( S_n \) of an arithmetic series is given by: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \] Substitute \( n = 25 \), \( a_1 = 53,000 \), and \( a_{25} = 91,400 \): \[ S_{25} = \frac{25}{2} \times (53,000 + 91,400) \] \[ S_{25} = \frac{25}{2} \times 144,400 \] \[ S_{25} = 12.5 \times 144,400 \] \[ S_{25} = 1,805,000 \] After 25 years, Marquis will have earned a total of $1,805,000 from this job.