A young couple plans to buy a property whose price is $450,000.$ They

have a deposit of $150,000,$ the rest will be covered by a mortgage, for

which the bank charges a fixed annual interest rate of $1\mathrm{.}7\%$ compounded

Amonthly. $=\frac{1}{12}$

$P=0\mathrm{.}017{}^n=25$

$P($i$)$ Assuming that they want to repay the mortgage in $25$ years, what

should be the monthly repayment?

Answer: The mortgage repayment is described by the equation

$D_{n+1}-D_n=r{D}_n-P,$

where $D_n$ is the debt after $n$ months, $P$ is the monthly repayment,

and $r$ is given by

$r=\frac{p}{100}$

where $p$ is the annual interest rate and $=\frac{1}{12}$ for an interest

compounded monthly. The solution is

please explain how the formula at the bottom is derived with step by step explanations. $D_n=(1+r{)}^n(D_0-\frac{P}{r})+\frac{P}{r}.$

where $D_0$ is the initial debt. Since $25$ years are $300$ months, the

required repayment is obtained by imposing $D_N=300=0$ and solving

for $P.$ We obtain

$P=\frac{r(1+r{)}^N{D}_0}{(1+r{)}^N-1}=1228\mathrm{.}2$