Answer:
$701.12
Step-by-step explanation:
Before doing the computing, notice that if we increase an amount D in,say, r%, it means that the new amount obtained is D+r% of D = D+(r/100)D = D(1+r/100).
That is to say, increasing an amount D in r% is equivalent to multiplying it by (1+r/100)
Taking into account that APR stands for Annual Percentage Rate, the amount we deposit will be increased 4% each year.
Generally, in this kind of loans the percentage is prorated monthly. That is to say, the money you have in the account will be increased in (4/12)%= 0.3333% each month.
Let D be the amount we are going to deposit each month.
After the month 1 we will have the money increased in 0.3333% plus the new deposit
[tex]D(1+\frac{0.3333}{100})+D=D(1+1.0033)[/tex]
After the month 2 we will have the money we already had increased in 0.3333% plus the new deposit D
[tex]D(1+1.0033)(1.0033)+D=D(1.0033+1.0033^2)+D=D(1+1.0033+1.0033^2)[/tex]
After the month 3 we will have, for the same reason,
[tex]D(1+1.0033+1.0033^2+1.0033^3)[/tex]
It can be noticed then, that after 18 years (96 moths) we will have an amount in the fund of
[tex]D(1+1.0033+1.0033^2+...+1.0033^{96})[/tex]
If we call
[tex]S=1+1.0033+1.0033^2+...+1.0033^{96}[/tex]
then
[tex]1.0033S=1.0033+1.0033^2+...+1.0033^{96}+1.0033^{97}[/tex]
Subtracting the equations
[tex]S-1.0033S=1-1.0033^{97}\Rightarrow S(1-1.0033)=1-1.0033^{97}[/tex]
and we have
[tex]S=\frac{1-1.0033^{97}}{1-1.0033}=114.10298[/tex]
So, after 18 years the amount in the fund will be
114.10298D
If we want this amount to be $80,000 then 114.10298D=80,000
[tex]D=\frac{80,000}{114.10298}\approx \$ 701.12[/tex]
So, the money we would have to deposit each month in a fund with an APR of 4% to accumulate $80,000 in 18 years, is
[tex]\boxed{D=\$701.12}[/tex]
