You just deposited $6,500 in a bank account that pays a 4.0% nominal interest rate, compounded quarterly. If you also add another $5,000 to the account one year (4 quarters) from now and another $7,500 to the account two years (8 quarters) from now, how much will be in the account three years (12 quarters) from now?



To find the total amount in the account three years from now with the additional deposits, we need to apply the formula for compound interest. Since the interest is compounded quarterly, we will calculate the future value for each deposit separately and then sum them up.

### Given:
- Initial deposit $ P_1 = 6,500 $
- Additional deposit after 1 year (4 quarters) $ P_2 = 5,000 $
- Additional deposit after 2 years (8 quarters) $ P_3 = 7,500 $
- Nominal interest rate $ r = 4.0\% $ per year
- Compounded quarterly, so the quarterly interest rate $ r_q = \frac{4.0\%}{4} = 1.0\% = 0.01 $
- Time period $ t = 3 $ years (or 12 quarters for the initial deposit)

### Formula:
The formula for compound interest is:

\[
A = P \times \left( 1 + \frac{r}{n} \right)^{n \cdot t}
\]

Where:
- $ P $ is the principal (initial deposit)
- $ r $ is the annual interest rate
- $ n $ is the number of times interest is compounded per year
- $ t $ is the time the money is invested for

For each deposit:
1. **Initial deposit ($ P_1 = 6,500 $) for 12 quarters:**
\[
A_1 = 6,500 \times (1 + 0.01)^{12}
\]

2. **Second deposit ($ P_2 = 5,000 $) for 8 quarters:**
\[
A_2 = 5,000 \times (1 + 0.01)^{8}
\]

3. **Third deposit ($ P_3 = 7,500 $) for 4 quarters:**
\[
A_3 = 7,500 \times (1 + 0.01)^{4}
\]

Let's calculate each amount.Here are the future values for each deposit:

1. The initial deposit of $6,500 grows to **$7,324.36** after 3 years.
2. The second deposit of $5,000 grows to **$5,414.28** after 2 years.
3. The third deposit of $7,500 grows to **$7,804.53** after 1 year.

The total amount in the account after 3 years is approximately **$20,543.18**.

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. What if the interest rate changes halfway through the period?
2. How would the result change if interest was compounded monthly instead of quarterly?
3. What happens if you add smaller deposits at more frequent intervals, like every quarter?
4. How does inflation affect the real value of your investment after 3 years?
5. How would the total change if the interest was compounded annually instead?

**Tip:** The frequency of compounding can significantly affect the growth of your investment!

You just deposited $6,500 in a bank account that pays a 4.0% nominal interest rate, compounded quarterly. If you also add another $5,000 to the account one year (4 quarters) from now and another $7,500 to the account two years (8 quarters) from now, how much will be in the account three years (12 quarters) from now?

Calculate how much a $6,500 deposit grows with a 4.0% nominal interest rate compounded quarterly. Additional deposits of $5,000 after 1 year and $7,500 after 2 years are included. Learn to apply the compound interest formula for different time periods to see how the total balance reaches approximately $20,543.18 after 3 years.

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