To determine the correct explicit formula and the balance at the beginning of year 5 for Zack's investment, let's analyze each given option.
1. Understanding the compound interest formula:
Compound interest calculated annually follows the formula:
[tex]\[
A = P \left(1 + \frac{r}{100}\right)^n
\][/tex]
Where:
- [tex]\(A\)[/tex] = the amount of money accumulated after n years, including interest.
- [tex]\(P\)[/tex] = the principal amount (the initial amount of money).
- [tex]\(r\)[/tex] = the annual interest rate (as a decimal).
- [tex]\(n\)[/tex] = the number of years the money is invested or borrowed for.
Given:
- Principal ([tex]\(P\)[/tex]) = \[tex]$700
- Annual interest rate (\(r\)) = 3% or 0.03 as a decimal.
- Number of years (\(n\)) = 5
Let's evaluate each option:
Option A: \(A(n) = 700 + (n-1)(0.03 \cdot 700)\)
- This formula represents a linear growth model, not compound interest.
- At \(n = 5\), this gives:
\[
A(5) = 700 + (5-1)(0.03 \cdot 700) = 700 + 4 \cdot 21 = 700 + 84 = 784
\]
- The balance is \$[/tex]784.00, which matches the given value but does not use the compound interest model.
Option B: [tex]\(A(n) = 700 + (0.003 \cdot 700)^{(n-1)}\)[/tex]
- This formula incorrectly represents the growth factor and exponentiation.
- At [tex]\(n = 5\)[/tex], this gives:
[tex]\[
A(5) = 700 + (0.003 \cdot 700)^{4} = 700 + 201.45 = 719.45
\][/tex]
- The balance is \[tex]$719.45, which does not reflect the compound interest model.
Option C: \(A(n) = 700 \cdot (1 + 0.03)^{(n-1)}\)
- This formula uses the correct compound interest concept but starts from year 1.
- At \(n = 5\), this gives:
\[
A(5) = 700 \cdot (1 + 0.03)^{4} = 700 \cdot 1.1255 = 787.86
\]
- The balance is \$[/tex]787.86, which is close but isn't the compound interest calculation for the end of year 5.
Option D: [tex]\(A(n) = 700 \cdot (1 + 0.03)^n\)[/tex]
- This formula correctly represents the compound interest calculation from the beginning.
- At [tex]\(n = 5\)[/tex], this gives:
[tex]\[
A(5) = 700 \cdot (1 + 0.03)^5 = 700 \cdot 1.159274 = 811.49
\][/tex]
- The balance is \[tex]$811.49, which accurately represents the compound interest after 5 full years.
Thus, the correct explicit formula to find the account’s balance at the beginning of year 5 is:
\[
\boxed{A(n) = 700 \cdot (1 + 0.03)^n}
\]
And the balance is:
\[
\boxed{\$[/tex]811.49}
\]
