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Simply wants to calculate interests of an initial amount $N$, with a monthly payback $m$, and a year interest rate of $R$

I guess the interests are not cumulated each month, but just at the end of the year.
Yet each month the total amount is decremented of 1 payback.
So it does $(N-6m)R$ interests at the end of the first year

2nd year initial amount: $N-12m+(N-6m)R$
2nd year interests: $(N-18m+(N-6m)R)R$

(... while the amount is positive)

I don't find any easy way to recurse it, if it's possible?

Edit

seeing comments, I think rather interests are added each month
so with $r=R/12$

1st month interests: $(Nr) +$
2nd month interests: $(N-m+Nr)r +$
3rd month interests: $( (N-m+Nr)(1+r) -m )r +$
4th month interests: $( (N-m+Nr)(1+r)^2 -m(1+r) -m )r +$
5th month interests: $( (N-m+Nr)(1+r)^3 -m(1+r)^2 -m(1+r) -m )r +$
6th month interests: $( (N-m+Nr)(1+r)^4 -m(1+r)^3 -m(1+r)^2 -m(1+r) -m )r +$
...
xth month interests: $( (N-m+Nr)(1+r)^x -m((1+r)^x-1)/r )r $
$= (Nr-m)(1+r)^{i+1} +m$
$= I(x)$

so the loan interests after x months could be calculated with:

$\sum_{1..x} \ I(y) = (x-1)m+(Nr-m)(1+r)^2((1+r)^x-1)/r$

but not sure if it's correct...

the yth month intial amount is $I(y)/r = 1/r*( (1+r)^{y+1} *(Nr-m) +m)$ that helps to know when the loan is finished

@Limitless I'm going to check if it gives same result with your answer

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    Yes, one can get recurrences, and explicit formulas. But the problem is not completely described. Is $r$ the nominal yearly rate? Is interest compounded monthly? What are we trying to calculate?2012-12-02
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    My answer attempts to interpret your answer as mathematically reasonable as possible, but I do not know if my interpretation of your question mirrors your thinking. In any event, I hope it is useful.2012-12-02
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    @AndréNicolas I thought when you do a loan, the interests were added at the end of the year only, but it's probably each months, it would increase slightly the result. yes r is the nominal interest rate (e.g. 2.5%), @ Limitless thx trying to understand it2012-12-02
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    Are you saying $r$ represents the interest paid to you for every year and $m$ is the amount taken out of the account every month?2012-12-02
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    yes r: yearly interests of the loan and m: monthly payback, @AndréNicolas I have edited the question with what you suggested can you have a look?2012-12-02
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    Though I do not know if your model is correct, I am pretty sure that our models are not the same. Mine is based on _yearly_ interest whereas yours is based on _monthly_ interest. This changes things.2012-12-02
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    I'm currently googling to know which option is commonly used, thx2012-12-02
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    @crl Is $R$ a nominal interest rate compounded monthly?2016-03-03
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    @crl Are payments and paybacks at the end or at the beginning of the month?2016-03-03

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