# loan repayment involving added amounts

A bank charges $5\%$ interest p.a on loan.

At the end of the year, the interest is added and then a fixed amount $R$ is paid off.

If the amount borrowed is $1000$, show that the amount owed at the start of year $n$ is given by:

$$(1000-20R)(1.05)^{n-1}+20R$$

Where does the "$20R$" come from?

You can think of it this way: If $A_n$ is the amount owned at the start of year $n$, you find $A_n=1.05A_{n-1}-R$, hence recursively \begin{aligned} A_n&=(1.05)^nA_0-R\sum_{k=0}^{n-1}(1.05)^k\\ &=(1.05)^nA_0-20R(1.05)^n+20R, \end{aligned} hence the "$20$" comes from dividing by $0.05$ in the formula for the geometric sum.