**Compound Interest** is the interest calculated on both the initial principal and the accumulated interest from previous periods. It differs from simple interest, which is calculated only on the original principal amount. Compound interest allows an investment or loan to grow at an accelerating rate over time, as interest is added to the principal periodically (annually, quarterly, monthly, etc.).
### Formula for Compound Interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- **A** is the amount of money accumulated after n years, including interest.
- **P** is the principal amount (initial investment).
- **r** is the annual interest rate (decimal).
- **n** is the number of times interest is compounded per year.
- **t** is the time the money is invested for in years.
### Example:
If you invest $1,000 at an annual interest rate of 5%, compounded annually for 3 years, the formula becomes:
\[
A = 1000 \left(1 + \frac{0.05}{1}\right)^{1 \times 3} = 1000 \times (1.05)^3 = 1000 \times 1.157625 = 1,157.63
\]
The compound interest earned would be $157.63.
### Key Points:
1. **Frequency of Compounding**: The more frequently interest is compounded, the higher the final amount. Common compounding periods are annually, quarterly, and monthly.
2. **Impact of Time**: The longer the time period, the more significant the impact of compound interest, especially with a higher interest rate.
Compound interest is a powerful tool for growing wealth, especially when the interest compounds over long periods, making it widely used in savings accounts, investments, and loans.
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