Complete the table and find the balance A if 

​$45004500 

is invested at an annual percentage rate of 

33​% 

for 10 years and compounded n times a year.


To complete the table and find the balance $ A $, we use the compound interest formula:

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

Where:
- $ P $ is the principal amount (\$4500)
- $ r $ is the annual interest rate (33% or 0.33)
- $ n $ is the number of times interest is compounded per year
- $ t $ is the time in years (10 years)
- $ A $ is the amount after time $ t $

Let's calculate $ A $ for different values of $ n $:

### 1. **$ n = 1 $ (Compounded Annually):**

\[
A = 4500 \left(1 + \frac{0.33}{1}\right)^{1 \times 10} = 4500 \left(1.33\right)^{10}
\]

### 2. **$ n = 4 $ (Compounded Quarterly):**

\[
A = 4500 \left(1 + \frac{0.33}{4}\right)^{4 \times 10} = 4500 \left(1 + 0.0825\right)^{40} = 4500 \left(1.0825\right)^{40}
\]

### 3. **$ n = 12 $ (Compounded Monthly):**

\[
A = 4500 \left(1 + \frac{0.33}{12}\right)^{12 \times 10} = 4500 \left(1 + 0.0275\right)^{120} = 4500 \left(1.0275\right)^{120}
\]

### 4. **$ n = 365 $ (Compounded Daily):**

\[
A = 4500 \left(1 + \frac{0.33}{365}\right)^{365 \times 10} = 4500 \left(1 + 0.0009041\right)^{3650} = 4500 \left(1.0009041\right)^{3650}
\]

Now, let's compute these values.

I'll proceed to calculate these balances.Here are the balances $ A $ for the different compounding frequencies:

1. **Compounded Annually ($ n = 1 $)**: $ A = \$77,934.35 $
2. **Compounded Quarterly ($ n = 4 $)**: $ A = \$107,233.06 $
3. **Compounded Monthly ($ n = 12 $)**: $ A = \$116,689.61 $
4. **Compounded Daily ($ n = 365 $)**: $ A = \$121,825.11 $

These values demonstrate how the frequency of compounding affects the final balance over time.

Do you want more details or have any questions? 

Here are five related questions to consider:
1. How would the balance change if the interest rate was lower?
2. What happens to the balance if the investment period is reduced to 5 years?
3. How does continuous compounding compare to daily compounding?
4. What is the effect of doubling the initial investment amount?
5. How would a semi-annual compounding frequency compare to the others?

**Tip:** The more frequently interest is compounded, the higher the final balance will be.

Explore how compound interest works with an initial investment of $4500 at an annual rate of 33% over 10 years, compounded annually, quarterly, monthly, and daily. Calculate the final balances and understand the impact of compounding frequency.

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