Oliver accepted a new job at a company with a contract guaranteeing annual raises of $2500. After working for the company for 8 years, Oliver will have a salary of $65000. Write an equation for S, in terms of n, comman, representing Oliver's salary after working nn years for the company.

To find the equation for Oliver's salary, let's break down the information step by step:

1. **Initial salary and raise pattern**:  
   Oliver receives an annual raise of $2500. This suggests that his salary increases in an arithmetic sequence, where the difference between consecutive terms is $2500.

2. **Salary after 8 years**:  
   After 8 years, Oliver's salary will be $65,000. We will use this to find his initial salary, which is the starting point of the arithmetic sequence.

### Step 1: Set up the formula for the arithmetic sequence
The general formula for an arithmetic sequence is:

\[
S = S_0 + d \cdot n
\]

Where:
- $ S $ is the salary after $ n $ years.
- $ S_0 $ is the initial salary.
- $ d $ is the common difference (annual raise), which is $2500.
- $ n $ is the number of years worked.

### Step 2: Use the information for year 8
From the information provided, we know that when $ n = 8 $, Oliver's salary is $65,000. Substituting into the equation:

\[
65,000 = S_0 + 2500 \times 8
\]

### Step 3: Solve for $ S_0 $
Now solve for the initial salary $ S_0 $:

\[
65,000 = S_0 + 20,000
\]

\[
S_0 = 65,000 - 20,000 = 45,000
\]

So, Oliver's initial salary was $45,000.

### Step 4: Write the equation for $ S $
Now that we know the initial salary, we can write the equation for Oliver's salary after $ n $ years:

\[
S = 45,000 + 2500 \cdot n
\]

This equation represents Oliver's salary after $ n $ years of working for the company.

---

Would you like further clarification or more details on this? Here are some related questions you might find helpful:

1. How would Oliver's salary look over the first 5 years?
2. If Oliver works for 12 years, what will his salary be?
3. How would the equation change if the annual raise was $3000 instead of $2500?
4. What if Oliver's starting salary was $50,000 instead of $45,000—how would that affect the equation?
5. If Oliver's salary increases only every 2 years, what would the new formula be?

**Tip:** Arithmetic sequences like this can be used to model many real-world situations involving constant growth or decline, such as savings or costs!

Oliver accepted a new job at a company with a contract guaranteeing annual raises of $2500. After working for the company for 8 years, Oliver will have a salary of $65000. Write an equation for S, in terms of n, representing Oliver's salary after working n years for the company.

Learn how to calculate Oliver's salary with an annual raise of $2500 using an arithmetic sequence. This step-by-step solution derives an equation to find his salary after n years of work, including finding the initial salary from a known value after 8 years. Perfect for students in Grades 9-11.

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