a. Use the appropriate formula to determine the periodic deposit. b. how
 much of the financial goal comes from deposits and how much comes from 
interest? Here is your data: Periodic deposit: $? at the end of each 
month, rate: 6.25% compounded monthly, time: 50 years, financial goal: 
1250000. Determine the periodic deposit. Do not round until the final 
answer. Then round up to the nearest dollar as needed.

### Part a: Finding the periodic deposit

To determine the periodic deposit for a future financial goal with compound interest, we use the **future value of an ordinary annuity formula**, which is as follows:

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

Where:
- $ FV $ is the future value (financial goal) = 1,250,000
- $ P $ is the periodic deposit (the value we need to find)
- $ r $ is the annual interest rate (6.25% = 0.0625)
- $ n $ is the number of times the interest is compounded per year (monthly, so $ n = 12 $)
- $ t $ is the number of years (50 years)

Let's substitute the known values into the formula and solve for $ P $:

\[
1,250,000 = P \times \frac{{(1 + 0.0625/12)^{12 \times 50} - 1}}{{0.0625/12}}
\]

We will calculate this step by step and solve for $ P $.

### Part b: Breakdown of deposits and interest

Once we find the periodic deposit $ P $, we can determine:
- The **total amount deposited**: $ P \times n \times t $
- The **total interest earned**: $ FV - $ (total amount deposited)

Let me calculate this now.### Part a: Periodic Deposit
The periodic deposit required to reach the financial goal of $1,250,000 is approximately **$302** (after rounding up to the nearest dollar).

### Part b: Breakdown of Deposits and Interest
- **Total amount deposited** over 50 years: \$181,046
- **Total interest earned**: \$1,068,954

This shows that most of the financial goal is achieved through interest rather than the deposits themselves.

Would you like more details on any part of the process?

Here are 5 related questions for further exploration:
1. How would the deposit change with a higher interest rate?
2. What happens if the deposits are made at the beginning of each month?
3. How would a shorter investment period (e.g., 30 years) affect the deposit?
4. What is the effect of compounding quarterly instead of monthly?
5. How much interest is earned in the first 10 years compared to the entire 50 years?

**Tip:** Small changes in the interest rate can have a significant impact over long periods due to compound interest.

Use the appropriate formula to determine the periodic deposit. How much of the financial goal comes from deposits and how much comes from interest? Here is your data. Periodic deposit: $? at the end of each month, rate: 6.25% compounded monthly, time: 50 years, financial goal: 1,250,000. Determine the periodic deposit. Do not round until the final answer. Then round up to the nearest dollar as needed.

Learn how to calculate the periodic deposit needed to achieve a financial goal of $1,250,000 over 50 years, with an interest rate of 6.25% compounded monthly. This problem involves financial mathematics concepts like future value and compound interest. Detailed step-by-step calculations included.

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