To model the amount of money in an investment account that compounds interest monthly and includes regular contributions, we can use a combination of both exponential growth from the interest and a linear addition from the monthly contributions.
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Compound Interest Calculation: The formula for compound interest is: \[ A = P(1 + r/n)^{nt} \] where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested or borrowed for, in years.
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Monthly Contributions: If $100 is contributed to the account at the end of each month, you can apply the future value of a series formula: \[ FV = C \frac{(1 + r/n)^{nt} - 1}{(r/n)} \] where \( C \) is the regular contribution (in this case, $100).
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Combined Model: The total amount in the account can be modeled as: \[ A(t) = P(1 + r/n)^{nt} + C \frac{(1 + r/n)^{nt} - 1}{(r/n)} \] This function depends on time \( t \) and is influenced by both the compounding interest on the initial amount and the contributions made over time.
Given these components, the overall growth of the investment account will primarily exhibit exponential behavior due to the compounding interest as well as the monthly contributions.
Since the growth is primarily driven by exponential factors, the best function to model the total amount in this scenario is:
Positive Exponential Function.