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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.

You have received a job offer with a starting salary of $45,000 per year with a guaranteed raise of $2,500 per year. How many years will it take before your total salary exceeds $500,000? What will the total gross salary be at the end of that year?

Find out how to calculate when your total salary exceeds $500,000 with a starting salary of $45,000 and a $2,500 raise each year. Includes linear growth, salary progression formula, and sum of arithmetic sequences.

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