To determine the balance of Paolo's investment at the beginning of year 5 with an annual compound interest rate of 4%, we'll begin by examining the correct explicit formula to use.
The correct formula for calculating the balance \( A(n) \) after \( n \) years when interest is compounded annually is:
[tex]\[ A(n) = P \cdot (1 + r)^n \][/tex]
where:
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate,
- \( n \) is the number of years.
Here:
- The principal amount \( P \) is \$500,
- The annual rate \( r \) is 4% or 0.04,
- The number of years \( n \) is 5.
Plugging these values into the formula, we get:
[tex]\[
A(n) = 500 \cdot (1 + 0.04)^5
\][/tex]
This simplifies to:
[tex]\[
A(5) = 500 \cdot (1.04)^5
\][/tex]
Now, we can compare this formula to the options provided:
A. \( A(n) = 500 + (0.004 \cdot 500)^{(n-1)} ; \$ 516.00 \)
B. \( A(n) = 500 \cdot (1 + 0.04)^n ; \$ 608.33 \)
C. \( A(n) = 500 + (n-1)(0.04 \cdot 500) ; \$ 580.00 \)
D. \( A(n) = 500 \cdot (1 + 0.04)^{(n-1)} ; \$ 584.93 \)
Option B matches the derived formula and explicitly uses the correct annual compound interest formula:
[tex]\[ A(n) = 500 \cdot (1 + 0.04)^n \][/tex]
At the beginning of year 5, substituting \( n = 5 \) into this formula, we find the calculation:
[tex]\[
A(5) = 500 \cdot (1.04)^5 \approx 608.33
\][/tex]
Therefore, the explicit formula and the balance at the beginning of year 5 are correctly given by option B:
[tex]\[ A(n) = 500 \cdot (1 + 0.04)^n \][/tex]
Thus, the balance at the beginning of year 5 is approximately \$608.33.
Summarizing:
The explicit formula is: \( A(n) = 500 \cdot (1 + 0.04)^n \)
The balance at the beginning of year 5 is: \$608.33.
