To determine the explicit formula for a compound interest geometric sequence, we need to recall the formula for compound interest:
[tex]\[ P_n = P_1 \cdot (1 + i)^{n-1} \][/tex]
Where:
- [tex]\( P_n \)[/tex] is the amount of money after [tex]\( n \)[/tex] periods.
- [tex]\( P_1 \)[/tex] is the initial amount of money (principal).
- [tex]\( i \)[/tex] is the interest rate per period.
- [tex]\( n \)[/tex] is the number of periods.
Let's compare the given choices with the standard compound interest formula:
A. [tex]\( P_n = P_1 \cdot (1+i)^{n-1} \)[/tex]
This matches our standard formula exactly.
B. [tex]\( P_n = P_1 \cdot (1-i)^{n+1} \)[/tex]
This formula improperly subtracts the interest rate and modifies the exponent incorrectly.
C. [tex]\( P_n = P_1 \cdot (1-i)^{n-1} \)[/tex]
This formula incorrectly subtracts the interest rate.
D. [tex]\( P_n = P_1 \cdot (1+i)^{n+1} \)[/tex]
This formula modifies the exponent incorrectly.
Given these comparisons, the explicit formula that correctly represents a compound interest geometric sequence is:
[tex]\[ A. \, P_n = P_1 \cdot (1+i)^{n-1} \][/tex]
So, the correct answer is:
A. [tex]\( P_n = P_1 \cdot (1+i)^{n-1} \)[/tex]
