monthly payment calculation?

Let's do a smaller loan repayment system and determine a general formula

(L * r - P) * r - P) * r - P = 0

L = loan amount

r = interest applied

P = payment

Solve for L

((Lr - P) * r - P) * r - P = 0

((Lr - P) * r - P) * r = P

(Lr - P) * r - P = P/r

(Lr - P) * r = P + P/r

Lr - P = P/r + P/r^2

Lr = P + P/r + P/r^2

L = P/r + P/r^2 + P/r^3

If we say 1/r = k, then we have

L = Pk + Pk^2 + Pk^3

L = P * (k + k^2 + k^3)

k + k^2 + k^3 is just a geometric sum. We can now generalize this problem to an n-number of payment

L = P * (k + k^2 + k^3 + ... + k^n)

S = k + k^2 + k^3 + ... + k^n

Multiply both sides by k

Sk = k^2 + k^3 + ... + k^(n + 1)

Sk - S = k^2 + k^3 + ... + k^(n + 1) - k - k^2 - k^3 - ... - k^n

S * (k - 1) = k^(n - 1) - k

S * (k - 1) = k * (k^(n) - 1)

S = k * (k^(n) - 1) / (k - 1)

L = P * S

L = P * k * (k^(n) - 1) / (k - 1)

L * (k - 1) / (k * (k^(n) - 1)) = P

P = L * (1/r - 1) / ((1/r) * ((1/r)^n - 1))

P = L * ((1 - r) / r) / ((1/r) * ((1/r)^n - 1))

P = L * (1 - r) / ((1/r)^n - 1)

r = 1 + i/a

i = annual interest rate

a = number of payments per year. In our case, a = 12 and i = 0.0765

r = 1 + 0.0765/12

r = 12.0765/12

P = L * (1 - r) / ((1/r)^n - 1)

P = L * (1 - 12.0765/12) / ((12/12.0765)^n - 1)

P = 531200 * (-0.0765/12) / ((12/12.0765)^n - 1)

P = 531200 * (0.0765/12) / (1 - (12/12.0765)^n)

n = 48 (12 payments per year for 4 years)

P = 531200 * (0.0765/12) / (1 - (12/12.0765)^48)

P = 12881.05023075757420766337881196102422823212147635184836445…

12881.050231 per month

For a total repayment of

618290.411076 (roughly)