geometric progression question?

Question

A bank loan of 500$ is arranged to be repaid in two years, by equal monthly installments,

interest, calculated monthly, is charged at 11% P.A On the remaining Debt,

calculate the monthly repayment if the first repayment is one month after the loan is made

the answer is 23.31 from the back of my text book,, how do i do it though!!,,,

Answer 1

I always write down the first several terms of these and hope to see a pattern:

Let's say
P is the principal = 500
r is the interest rate = .11
n is the number of months
M is the monthly repayment (that we're looking for)

The amount of money to repay at the end of month n is:
n=0 => P
n=1 => P(1+r/12) - M
n=2 => (P(1+r/12) - M) (1+r/12) - M
n=3 => ((P(1+r/12) - M) (1+r/12) - M) (1+r/12) - M

Now collect terms carefully:
n => P(1+r/12)^n -M -M(1+r/12) -M(1+r/12)^2 - ... -M(1+r/12)^(n-1)
and since the second part is a geometric progression it simplifies to (and we remember that we are really looking to solve for M at the end):
= P(1+r/12)^n -M[(1+r/12)^n-1]/((1+r/12)-1)
= P(1+r/12)^n - M[(1+r/12)^n-1]/(r/12)
= P(1+r/12)^n - 12M[(1+r/12)^n-1]/r

Now if you should repay this after n=24 months, then at that time the above amount will be 0:
12M[(1+r/12)^n-1]/r = P(1+r/12)^n
M[(1+r/12)^n-1] = (Pr/12)(1+r/12)^n
M = (Pr/12)(1+r/12)^n / [(1+r/12)^n-1]
M = (Pr/12) / [1 - (1+r/12)^(-n)]

Now just plug in, and you'll get your answer

Answer 2

First search the monthly interest rate
(1+x)^12 = 1.11
so 12log(1+x) =log 1,11 and log (1+x) =log1.11/12 and 1+x=1.0087
500= (1.0087 +1.0087 ^2++++++++1.0087^24)*Monthly Pay
500 = (1.0087^25-1.0087)/0.0087 *Monthly pay
500=26.7927* Monthly pay
Monthly pay =$18.66