To create a formula for Carly's savings over the course of time, we need to define a few variables and establish the relationship between them. Let's assume the following:
- Let \( S(t) \) be Carly's total savings at time \( t \).
- Let \( P \) be the principal amount or the initial amount Carly starts with.
- Let \( r \) be the annual interest rate (as a decimal).
- Let \( t \) be the time in years.
- Let \( C \) be the amount Carly saves or contributes to her savings each year.
Step 1: Identify the Components of the Formula
- Initial Savings: This is the principal amount \( P \).
- Annual Contributions: This would be a fixed amount \( C \) that Carly adds to her savings each year.
- Interest Accumulation: The interest on the total savings can be modeled using compound interest.
Step 2: Develop the Formula
For simplicity, let's assume Carly's savings grow with compound interest, and that contributions are made at the end of each year.
The formula for the total amount accumulated after \( t \) years with an initial amount \( P \) and annual contributions \( C \) is given by:
\[ S(t) = P(1 + r)^t + C \left( \frac{(1 + r)^t - 1}{r} \right) \]
Explanation of the Formula Components
- First Component: \( P(1 + r)^t \) represents the growth of the initial savings due to compound interest over \( t \) years.
- Second Component: \( C \left( \frac{(1 + r)^t - 1}{r} \right) \) accounts for the contributions made each year at the end of the year, which also grow with interest.
Final Formula
Thus, the final formula representing Carly's savings over time is:
\[ S(t) = P(1 + r)^t + C \left( \frac{(1 + r)^t - 1}{r} \right) \]
Example
If Carly starts with an initial savings of $1,000 (P = 1000), contributes $500 every year (C = 500), and the annual interest rate is 5% (r = 0.05), then her savings after \( t \) years would be calculated as follows:
\[ S(t) = 1000(1 + 0.05)^t + 500 \left( \frac{(1 + 0.05)^t - 1}{0.05} \right) \]
This formula can be used to estimate Carly's savings at any point in time.
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