Compounding interest

Definition

Compounding interest is the process where the value of an investment increases because the earnings on an investment, both capital gains and interest, earn interest as time passes. This exponential growth can be modeled using functions involving exponents.

5 Must Know Facts For Your Next Test

1. Compounding interest can be represented by the formula $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the amount of money accumulated, $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times that interest is compounded per year, and $t$ is the time in years.
2. The base of the exponential function in compounding interest formulas involves $(1 + \frac{r}{n})$, showing an increment over 1 due to interest.
3. Continuous compounding uses Euler's number $e$ and is represented by the formula $A = Pe^{rt}$.
4. Exponential growth in compounding interest demonstrates how small changes in interest rates or compounding frequency can significantly impact final amounts over time.
5. Logarithms are used to solve for variables within compounding interest equations when any parameters other than time need to be determined.

Review Questions

• What is the general formula for compound interest?
• How does continuous compounding differ from periodic compounding?
• Why are logarithms useful in solving compound interest problems?

Related terms

Exponential Function: A mathematical function of the form $f(x) = a \cdot b^x$, where $b > 0$ and $b \neq 1$, describing growth or decay.

Logarithmic Function: The inverse function of an exponential function, often expressed as $y = \log_b(x)$.

$e$ (Euler's Number): $e$ is an irrational number approximately equal to 2.71828 and serves as the base for natural logarithms.