Mathematical deep-dive: how it works

We consider fixed monthly payments m.

for a number of months N

at a rate r.

We want to determine the total amount borrowed b

as well as the monthly interest .

Observe that for the k-th month, the interests are as follows:

Another way to see this interest is by checking the decrement, that is by how much the interest amount decrease from a month to the next one:

Equivalently, that is:

Note that this sequence is arithmetico-geometric.

Let us pose the new sequence such that:

Then, rearranging the terms we have:

Thus a is a geometric sequence. We can thus express it directly:

We can now go back to the interest amounts:

Let us denote the associated series with the following:

Then we have:

Since the sum of all monthly payments precisely equals the sum of the total amount borrowed plus the sum of all interests, We know that:

We can then deduce b:

With a little bit of manipulation, we arrive at the final formula:

QED