Sure, let's complete the table and find the balances for each compounding frequency.
1. Principal (P): The initial amount invested is [tex]$3500.
2. Annual interest rate (r): 3% or 0.03 as a decimal.
3. Time (t): 10 years.
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Let's calculate the balances for different values of \( n \):
### When \( n = 1 \) (Compounded Annually)
\[ A = 3500 \left(1 + \frac{0.03}{1}\right)^{1 \cdot 10} \]
\[ A = 3500 \left(1 + 0.03\right)^{10} \]
\[ A = 3500 \left(1.03\right)^{10} \]
\[ A \approx 4703.71 \]
### When \( n = 2 \) (Compounded Semiannually)
\[ A = 3500 \left(1 + \frac{0.03}{2}\right)^{2 \cdot 10} \]
\[ A = 3500 \left(1 + 0.015\right)^{20} \]
\[ A = 3500 \left(1.015\right)^{20} \]
\[ A \approx 4713.99 \]
### When \( n = 4 \) (Compounded Quarterly)
\[ A = 3500 \left(1 + \frac{0.03}{4}\right)^{4 \cdot 10} \]
\[ A = 3500 \left(1 + 0.0075\right)^{40} \]
\[ A = 3500 \left(1.0075\right)^{40} \]
\[ A \approx 4719.22 \]
### When \( n = 12 \) (Compounded Monthly)
\[ A = 3500 \left(1 + \frac{0.03}{12}\right)^{12 \cdot 10} \]
\[ A = 3500 \left(1 + 0.0025\right)^{120} \]
\[ A = 3500 \left(1.0025\right)^{120} \]
\[ A \approx 4722.74 \]
### When \( n = 365 \) (Compounded Daily)
\[ A = 3500 \left(1 + \frac{0.03}{365}\right)^{365 \cdot 10} \]
\[ A = 3500 \left(1 + 0.00008219\right)^{3650} \]
\[ A = 3500 \left(1.00008219\right)^{3650} \]
\[ A \approx 4724.45 \]
Now, let's fill in the table with these balances:
\[
\begin{tabular}{c|l|l|l|l}
\hline
n & 1 & 2 & 4 & 12 & 365 \\
\hline
A & 4703.71 & 4713.99 & 4719.22 & 4722.74 & 4724.45 \\
\hline
\end{tabular}
\]
So, the balance \( A \) after 10 years for each compounding frequency \( n \) is:
- For annual compounding (\( n = 1 \)): $[/tex]4703.71
- For semiannual compounding ([tex]\( n = 2 \)[/tex]): [tex]$4713.99
- For quarterly compounding (\( n = 4 \)): $[/tex]4719.22
- For monthly compounding ([tex]\( n = 12 \)[/tex]): [tex]$4722.74
- For daily compounding (\( n = 365 \)): $[/tex]4724.45
