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Let's analyze the problem step by step.

### Given Information:
- Initial annual salary: $ S_0 = 56,345 $ dollars
- Annual pay raise: $ 3.45\% $ per year
- We need to find the salary after 11 years, using a recurrence relation.

### Step 1: Set up the recurrence relation
Let $ S_n $ represent the annual salary after $ n $ years. The recurrence relation for the salary increase can be expressed as:
\[ S_{n+1} = S_n \times (1 + 0.0345) \]
where $ S_0 = 56,345 $.

### Step 2: Calculate the annual salary after 11 years
To find the salary after 11 years, we will use the formula:
\[ S_{11} = S_0 \times (1 + 0.0345)^{11} \]

Substituting the values:
\[ S_{11} = 56,345 \times (1.0345)^{11} \]

Let's calculate this value:

1. Compute $ (1.0345)^{11} $
2. Multiply the result by 56,345
3. Round the final answer to the nearest whole dollar.

### Calculation:
1. $ (1.0345)^{11} \approx 1.457392 $ (approximately)
2. $ S_{11} = 56,345 \times 1.457392 \approx 82,118.94 $

Rounding to the nearest whole dollar:
\[ S_{11} \approx 82,119 \text{ dollars} \]

### Final Answer
The annual salary after 11 years will be **$82,119**.

Would you like further details or have any questions on this solution? Here are a few related questions for further exploration:

1. How would the recurrence relation change if the annual pay raise was different, say 5%?
2. What is the effect of compounding in recurrence relations like these?
3. How would the answer differ if we calculated the salary over a shorter time span, such as 5 years?
4. How can we express this problem using a direct (non-recursive) formula?
5. What would be the future salary if you started with a different initial amount, for example, $60,000?

**Tip:** Understanding recurrence relations is useful not just in finance, but in many areas like population growth, computer algorithms, and physics!

Let's say that you accept a job that pays $56,345 annually. You receive a 3.45% pay raise each year. Set up a recurrence relation that models your annual pay. What will your annual pay be after 11 years?

Learn how to model an annual salary increase using a recurrence relation. In this example, starting with $56,345 and a 3.45% annual raise, we calculate the salary after 11 years. Understand key concepts such as recurrence relations, percentage increases, and exponential growth.

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