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.

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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!