To determine which options can be represented by the expression \( A = P(1 + r/n)^{nt} \), where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial balance).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
Let's analyze each option:
A. The balance of a savings account after \( t \) years if the initial balance is $500 and it is invested at a 3.5% annual interest rate compounded annually.
- Here, \( P = 500 \), \( r = 0.035 \), \( n = 1 \) (compounded once a year).
- This can be represented as: \[ A = 500(1 + 0.035/1)^{1 \cdot t} \]
B. The balance of a savings account after \( t \) years if the initial balance is $500 and it is invested at a 0.875% annual interest rate compounded annually.
- Here, \( P = 500 \), \( r = 0.00875 \), \( n = 1 \).
- This can be represented as: \[ A = 500(1 + 0.00875/1)^{1 \cdot t} \]
C. The balance of a savings account after \( t \) years if the initial balance is $500 and it is invested at a 0.875% annual interest rate compounded quarterly.
- Here, \( P = 500 \), \( r = 0.00875 \), \( n = 4 \) (because it is compounded quarterly).
- This can be represented as: \[ A = 500(1 + 0.00875/4)^{4t} \]
D. The balance of a savings account after \( t \) years if the initial balance is $500 and it is invested at a 3.5% annual interest rate compounded quarterly.
- Here, \( P = 500 \), \( r = 0.035 \), \( n = 4 \).
- This can be represented as: \[ A = 500(1 + 0.035/4)^{4t} \]
Based on the analysis:
- Option A can be represented by the formula.
- Option B can be represented by the formula.
- Option C can be represented by the formula.
- Option D can be represented by the formula.
Thus, all four options (A, B, C, and D) can be represented by the expression given.
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