To express an exponential function for compounding interest, we can start with the general formula:
A = P(1 + r/n)^(nt)
In this formula:
- 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.
Given that the rate of change is 5, we interpret this as the annual interest rate r being 5% or 0.05 in decimal form. Assuming the interest is compounded once per year (n = 1), the function simplifies to:
A = P(1 + 0.05)^t
This function indicates that the amount A grows exponentially with time t due to the compounding effect of interest.
For example, if you invest $1000 at an interest rate of 5% compounded annually, after 1 year you would have:
A = 1000(1 + 0.05)^1 = 1000(1.05) = $1050
After 2 years:
A = 1000(1 + 0.05)^2 = 1000(1.05)^2 = 1000(1.1025) = $1102.50
This shows how the investment grows over time, illustrating the effect of compounding interest.