The problem asks about a job offer with a starting salary of $45,000 per year, along with a guaranteed raise of $2,500 each year. There are two main questions to answer:

1. How many years will it take before your total salary exceeds $500,000?
2. What will your total gross salary be at the end of that year?

Solution Steps:

1. Understanding the Salary Progression:

Each year, your salary increases by $2,500. This means that your salary follows a linear growth pattern.

Let’s denote:

• $S_n$ as the salary in the $n$-th year.
• Starting salary $S_1 = 45,000$.
• The salary increases by $2,500 each year.

Thus, the salary in the $n$-th year can be modeled as: $S_n = 45000 + (n-1) \times 2500$ where $n$ is the number of years.

2. Cumulative Salary:

To find when your total salary exceeds $500,000, we need to compute the sum of salaries over multiple years. The total salary after $n$ years is the sum of an arithmetic sequence: $\text{Total salary after } n \text{ years} = S_1 + S_2 + \dots + S_n$

The sum of the first $n$ terms of an arithmetic sequence is given by: $\text{Total salary} = \frac{n}{2} \times (S_1 + S_n)$

We want to solve for $n$ such that the total salary exceeds $500,000.

Let me calculate this for you.It will take 10 years for your total salary to exceed $500,000.

At the end of the 10th year:

• Your salary will be $67,500.
• The total gross salary after 10 years will be $562,500.

Would you like more details, or do you have any questions?

Here are 5 related questions to explore:

1. How would the calculations change if the raise was compounded instead of fixed?
2. What if the starting salary was different, say $50,000 instead of $45,000?
3. How would the total salary change if you skipped raises for two years?
4. What is the effect on total earnings if the raise increases each year (e.g., by $500 per year)?
5. How long would it take to reach a total salary of $1,000,000?

Tip: The formula for summing an arithmetic sequence is helpful in solving problems where quantities increase linearly, like in salary-related problems.