SOLUTION: Please help me to solve this problem: Mark takes out a loan of 40,000 that is must be repaid over the course of 5 years. Interest is compounded at a nominal rate of 9% convertib

Question 949625: Please help me to solve this problem:
Mark takes out a loan of 40,000 that is must be repaid over the course of 5 years. Interest is compounded at a nominal rate of 9% convertible monthly and he makes 60 scheduled repayments of 735 at the end of each month. What additional payment must he make at the end of the term so that the loan is completely paid off?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
nominal rate convertible monthly means that
each of the year (1 month) they add of ,
that is times the old balance,
to the old balance to calculate the new balance.

At the end of the first month, just as Mark is about to make his first payment,
they add to the to get
,
which is times the previous balance.
Immediately, Mark makes his payment, and the balance goes down to
.

At the end of the second month, just as Mark is about to make his second payment,
the balance again increases by a factor of ,
from to ,
only to get immediately reduced by the second payment
to .

At the end of the second month,
the balance first increases again by a factor of ,
to , and then
decreases to
with the third payment.

The pattern repeats each month, so after months, and payments, the balance is

=
From either geometric sequences and series,
or from polynomial s and factoring, we know that
.
So the final balance, after months, and payments is

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