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The problem is about calculating the future value of an annuity and additional inheritance compounded over time.

Let's break it down:

1. **Part (a)**: Nick has been making monthly deposits of $110 for 35 years (or 420 months), with a 7% annual interest rate compounded monthly. 

   The formula to calculate the future value of an annuity is:

   \[
   A = P \times \frac{(1 + r)^n - 1}{r}
   \]

   where:
   - $ P $ is the monthly deposit ($110),
   - $ r $ is the monthly interest rate (7% annual rate means $ \frac{7}{12 \times 100} = 0.0058333 $ per month),
   - $ n $ is the total number of months (35 years = 420 months).

2. **Part (b)**: Nick received an inheritance of $15,000 10 years ago, which he deposited into the same retirement fund. We need to find the future value of this lump sum after 10 years with the same 7% interest rate compounded monthly.

   The formula to calculate the future value of a lump sum is:

   \[
   A = P \times (1 + r)^n
   \]

   where:
   - $ P $ is the lump sum ($15,000),
   - $ r $ is the monthly interest rate (0.0058333),
   - $ n $ is the total number of months (10 years = 120 months).

The answers provided are:
- $ \text{Part (a)}: \, A = 198,116.01 $
- $ \text{Part (b)}: \, A = 27,321.83 $

Do you need further explanation on the calculations? If so, let me know!

Here are some follow-up questions:
1. How does increasing the interest rate affect the accumulated amount in an annuity?
2. What happens to the future value if Nick deposited $150 instead of $110 monthly?
3. How would the calculations change if the interest was compounded quarterly instead of monthly?
4. How would delaying the inheritance deposit by 5 years impact the future value in Part (b)?
5. What formula would you use if the interest was compounded continuously?

**Tip**: Always break compound interest problems into smaller steps to ensure precision!

Upon graduating from college 35 years ago, Dr. Nick Riviera has been making monthly deposits into a retirement fund. His final deposit is $110 per month. The fund has earned 7% interest compounded monthly. How much has Nick accumulated in his retirement account after 35 years? Additionally, 10 years ago, Nick received an inheritance of $15,000, which he deposited in the retirement account. What is the current balance in his account?

This problem calculates the future value of an annuity using monthly deposits of $110 over 35 years with a 7% annual interest rate, compounded monthly. It also explores how a $15,000 lump sum inheritance grows over 10 years in the same account. Learn how to use annuity and compound interest formulas to solve real-world retirement problems.

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