To derive the explicit formula that can be used to find the account's balance at the beginning of year 7, let's recall how compound interest works. The general formula for compound interest is given by:
[tex]\[ A(t) = P \times (1 + r)^t \][/tex]
Where:
- [tex]\( A(t) \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
In this specific problem:
- The principal amount [tex]\( P \)[/tex] is \$800.
- The annual interest rate [tex]\( r \)[/tex] is 3%, which is 0.03 in decimal form.
- The time [tex]\( t \)[/tex] is 7 years.
By substituting the given values into the compound interest formula, we get:
[tex]\[ A(7) = 800 \times (1 + 0.03)^7 \][/tex]
Thus, the explicit formula that can be used to find the account's balance at the beginning of year 7 is:
[tex]\[ A(7) = 800 \times (1 + 0.03)^7 \][/tex]
This corresponds to option C in the given choices:
[tex]\[ A(7) = 800 \cdot(1+0.03)^7 \][/tex]
Therefore, the correct answer is:
C. [tex]\( A(7)=800 \cdot(1+0.03)^7 \)[/tex]
