Consider a bank account in which $P_o$ is invested at the beginning of a compounding period, with a nominal interest rate $r$ and compounding $K$ times per year $($so each compounding period is $(\frac{1}{K}{)}^{th}$ of one year$).$ How much will be in the account after $N$ compounding periods? Let $P_j$ denote the balance in the account after $j$ compounding periods, including the interest earned in the last of these $j$ periods. The balance at the end of a new compounding period is the balance at the end of the proceeding period plus the interest that the older balance earned during compounding period:

$P_{j+1}=P_j+$ the interest earned by $P_j$ in one compounding period

$=P_j+($nominal rate