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? $$ \$ 1,792,000 $$ $$ \$ 1,805,000 $$ $$ \$ 1,844,975 $$ $$ \$ 1,845,000 $$

Question

UpStudy AI · Accepted Answer

To determine the total amount Marquis will have earned after 25 years, let's break down the problem step by step.

### Given:
- **Starting Salary:** \$53,000 per year
- **Annual Raise:** \$1,600 per year
- **Number of Years:** 25

### Salary for Each Year:
The salary for each year can be modeled by the equation:
$$
a_n = 53,000 + (n - 1) \times 1,600
$$
where:
- $$ a_n $$ is the salary in the $$ n^{th} $$ year.
- $$ n $$ is the number of years.

### Total Earnings Over 25 Years:
To find the total earnings over 25 years, we need to sum the salaries from year 1 to year 25.

$$
\text{Total Earnings} = \sum_{n=1}^{25} a_n = \sum_{n=1}^{25} \left(53,000 + (n - 1) \times 1,600\right)
$$

This can be broken down into two separate sums:
1. **Constant Salary Component:**
$$
\sum_{n=1}^{25} 53,000 = 25 \times 53,000 = 1,325,000
$$

2. **Raise Component:**
$$
\sum_{n=1}^{25} (n - 1) \times 1,600 = 1,600 \times \sum_{n=0}^{24} n = 1,600 \times \frac{24 \times 25}{2} = 1,600 \times 300 = 480,000
$$

### Adding Both Components:
$$
\text{Total Earnings} = 1,325,000 + 480,000 = 1,805,000
$$

### **Answer:**
$$\$1,805,000$$
After 25 years, Marquis will have earned a total of \$1,805,000.

Marquis begins a job making per year. Each year, he is to receive a raise. His salary after years can
be modeled by this equation: . After 25 years, how much total money will Marquis have
earned from this job?

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Solución

Given:

Salary for Each Year:

Total Earnings Over 25 Years:

Adding Both Components:

Answer:

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Marquis begins a job making per year. Each year, he is to receive a raise. His salary after years can be modeled by this equation: . After 25 years, how much total money will Marquis have earned from this job?