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Which of the following is the explicit formula for a compound interest geometric sequence?

A. [tex]P_n = P_1 \bullet (1+i)^{n-1}[/tex]
B. [tex]P_n = P_1 \cdot (1-i)^{n+1}[/tex]
C. [tex]P_n = P_1 \bullet (1-i)^{n-1}[/tex]
D. [tex]P_n = P_1 \bullet (1+i)^{n+1}[/tex]



Answer :

GinnyAnswer
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]

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