How is mortgage monthly payment calculated?

Starting with a loan $L$ and annual interest rate $a$, let’s list the amounts of debt we owe the bank for each month.

To simplify the formulas, let’s write $r = 1 + a/12$, this is the rate at which your is increasing monthly. And $P$ is the monthly payment you are supposed to make to the bank.

Month	Debt
$$0$$	$$L$$
$$1$$	$$Lr - P$$
$$2$$	$$(Lr - P)r - P \\= Lr^2 - Pr - P$$
$$3$$	$$(Lr^2 - Pr - P)r - P \\= Lr^3 - Pr^2 - Pr - P$$
$$\cdots$$	$$\cdots$$
$$n$$	$$Lr^n - Pr^{n-1} - Pr^{n-2} - \cdots - P \\ =$$

At the nth monthly your debt should be cleared, so, setting it to 0 we get:

$$\begin{align*} P &= \frac{L(r - 1)r^n}{r^n - 1} \\ &= \frac{L\frac{a}{12}(1 + \frac{a}{12})^n}{(1 + \frac{a}{12})^n - 1} \end{align*}$$