A principal of ​$5000 is invested in an account paying an annual rate of 7​%. Find the amount in the account after 6 years if the account is compounded​ semiannually, quarterly, and monthly.

To solve this problem, we can use the compound interest formula:

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

where:
- $A$ is the amount of money in the account after $t$ years,
- $P$ is the principal amount (initial investment),
- $r$ is the annual interest rate (in decimal form),
- $n$ is the number of times the interest is compounded per year,
- $t$ is the number of years the money is invested or borrowed for.

Given:
- $P = 5000$ dollars,
- $r = 7\%$ or $0.07$ in decimal,
- $t = 6$ years.

We need to find the amount $A$ for three cases:
1. Semiannually ($n = 2$),
2. Quarterly ($n = 4$),
3. Monthly ($n = 12$).

### 1. Compounded Semiannually

\[
A = 5000 \left(1 + \frac{0.07}{2}\right)^{2 \times 6}
\]

### 2. Compounded Quarterly

\[
A = 5000 \left(1 + \frac{0.07}{4}\right)^{4 \times 6}
\]

### 3. Compounded Monthly

\[
A = 5000 \left(1 + \frac{0.07}{12}\right)^{12 \times 6}
\]

Let me calculate these amounts.The amounts in the account after 6 years for different compounding periods are as follows:

1. **Compounded Semiannually**: $7,555.34
2. **Compounded Quarterly**: $7,582.21
3. **Compounded Monthly**: $7,600.53

These differences arise because more frequent compounding leads to slightly more interest being accrued over time.

Would you like more details on any of these calculations, or do you have any other questions?

Here are five related questions:
1. How would the final amount change if the interest rate were 8% instead of 7%?
2. What is the impact of increasing the number of compounding periods (e.g., daily compounding)?
3. How would the amounts compare if the investment period was 10 years instead of 6?
4. Can you calculate the effective annual rate (EAR) for each compounding period?
5. What is the total interest earned in each case?

**Tip:** The more frequently interest is compounded, the greater the final amount will be, due to the effect of interest on interest.

Explore how compound interest accumulates over 6 years with a principal of $5000 and an annual interest rate of 7%, compounded semiannually, quarterly, and monthly. Get detailed calculations and understand the impact of compounding frequency on final amounts.

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