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!