Deduction of calculation formula of equal and principal interest

where P is full, r is monthly, and N is the number of periods.

For equal principal and interest, the repayment amount of each month is the same, but the interests are decreasing, the first month is the full one-month interest, the second month is to remove the first month after the principal amount of the remaining one months interest, as follows:

Assuming a monthly repayment of X, full m, monthly R

A1 = X-m*r
A2 = x-(M-A1) *RA3 = x-(M-A1-A2) *r
...
an = X-(M-A1-...-an-1) *r
an+1 = x (because A1 + ... + an=m)

Substituting the A1 into the A2 expression, and then substituting A1 and A2 into the A3 expression, you get
A3 = (X-MR) * (1+R) 2
a4 = (X-MR) * (1+R) 3

...
An+1 = (X-MR) * (1+r) n = x
This results in a formula for the X of equal and principal interest.

Deduction of calculation formula of equal and principal interest