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Bot GPT 4-o mini To express the amount $15,528.23 in the form of a formula for the future value of an annuity due to monthly deposits compounded at a specified interest rate, we can use the formula for the future value of a series of annuity payments.
The future value of an annuity can be calculated using the formula:
\[
FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
Where:
- \( FV \) is the future value of the annuity,
- \( P \) is the amount of each payment (in this case, the monthly deposit),
- \( r \) is the monthly interest rate (annual rate divided by number of compounding periods),
- \( n \) is the total number of payments (number of years times number of payments per year).
In this case:
- \( P = 100 \) (the monthly deposit),
- The annual interest rate is \( 5\% \), so the monthly interest rate \( r = \frac{5\%}{12} = \frac{0.05}{12} \approx 0.0041667 \),
- \( n = 10 \text{ years} \times 12 \text{ months/year} = 120 \text{ months} \).
Now, substituting these values into the formula gives:
\[
FV = 100 \times \left( \frac{(1 + 0.0041667)^{120} - 1}{0.0041667} \right)
\]
Calculating \( (1 + 0.0041667)^{120} \):
\[
= (1.0041667)^{120} \approx 1.647009
\]
Now, plug this into the equation:
\[
FV = 100 \times \left( \frac{1.647009 - 1}{0.0041667} \right)
\]
\[
= 100 \times \left( \frac{0.647009}{0.0041667} \right)
\]
\[
\approx 100 \times 155.2811 \approx 15528.11
\]
Rounding \( 15528.11 \) gives approximately \( 15528 \). However, since we want to express $15,528.23 as per your request, we can see if there were adjustments to make throughout this process.
To express the amount \( 15,528.23 \) under the form of the future value formula:
So we retain the \( FV \) value as \( 15,528.23 \) and keep it contingent on the aforementioned \( P \), \( r \), and \( n \):
\[
15,528.23 = 100 \times \left( \frac{(1 + 0.00416667)^{120} - 1}{0.00416667} \right)
\]
Thus, the amount of $15,528.23 can be expressed in the form of this annuity future value formula.