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img of Marquis begins a job making $53,000 per year. Each year, he...

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 n years can be modeled by this equation: an=53,000+(n1)1,600 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 n years is given by the equation: an=53,000+(n1)×1,600 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: a1=53,000 a_1 = 53,000 Next, calculate the salary for year 25: a25=53,000+(251)×1,600 a_{25} = 53,000 + (25 - 1) \times 1,600 a25=53,000+24×1,600 a_{25} = 53,000 + 24 \times 1,600 a25=53,000+38,400 a_{25} = 53,000 + 38,400 a25=91,400 a_{25} = 91,400 The total money earned is the sum of an arithmetic series from year 1 to year 25. The sum Sn S_n of an arithmetic series is given by: Sn=n2×(a1+an) S_n = \frac{n}{2} \times (a_1 + a_n) Substitute n=25 n = 25 , a1=53,000 a_1 = 53,000 , and a25=91,400 a_{25} = 91,400 : S25=252×(53,000+91,400) S_{25} = \frac{25}{2} \times (53,000 + 91,400) S25=252×144,400 S_{25} = \frac{25}{2} \times 144,400 S25=12.5×144,400 S_{25} = 12.5 \times 144,400 S25=1,805,000 S_{25} = 1,805,000 After 25 years, Marquis will have earned a total of $1,805,000 from this job.

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