# Difference between revisions of "022 Sample Final A, Problem 10"

$A\,=\,P\left(1+{\frac {r}{n}}\right)^{nt},$ where $A$ is the value of the account, $P$ is the principal (original amount invested), $r$ is the annual rate and $n$ is the number of compoundings per year. The value of $n$ is $365$ for compounding daily, $52$ for compounding weekly, and $12$ for compounding monthly. As a result, the exponent $nt$ , where $t$ is the time in years, is the number of compounding periods where we actually earn interest. Similarly, $r/n$ is the rate per compounding period (the annual rate divided by the number of compoundings per year).
For example, if we compound monthly for $7$ years at a $6\%$ rate, we would compound $nt\,=\,12\cdot 7\,=\,84$ times, once per month, at a rate of $0.06/12\,=\,0.005$ per monthly period. Notice that we always use the decimal version for interest rates when using these equations.
We are given all the pieces required. We begin with $\2100$ of principal, and compound quarterly, or $4$ times per year. Using the formula in 'Foundations', the equation for the account value after 8 years is
$A\,=\,P\left(1+{\frac {r}{n}}\right)^{nt}\,=\,2100\left(1+{\frac {0.06}{4}}\right)^{4\cdot 8}\,=\,2100(1.015)^{32}.$ $A\,=\,2100(1.015)^{32}.$ 