## Continuously Compounded Interest

The value of an investment or deposit into a savings account increases depending on the interest rate and how often that interest is added the account balance. For an initial investment of {eq}P {/eq} and an interest rate {eq}r {/eq}, the amount after {eq}t {/eq} years when this interest is compounded continuously can be found by calculating {eq}A = Pe^{rt} {/eq}.

The future value of an investment or bank account is the value this amount grows to after time has passed. The initial value for this problem is 20,000, which was placed into an account with an interest rate of 3.8% per year. The value of this account after 15 years can be found as follows. {eq}\begin{align*} A &= 20000e^{0.038(15)}\\ &= 20000e^{0.57}\\ &= 20000(1.76827)\\ &= 35365.4 \end{align*} {/eq} Thus, the future value of this investment is35,365.40.