Financial mathematics - annuities and loans please help

[mellow juniper]

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[buoyant veldt]

Let's try solving this generally.
Suppose the initial amount is N0, the rate each month is p and the payment each month is n.
I assume the interest happens before the payment. Then we have:
N(t + 1) = (1 + p)N(t) - n, N(0) = N0
Solving this recurrence relation gives:
N(t) = (n - (n - pN(0))(1 + p)^t)/p
Suppose the total payment period is T. Then:
N(T) = (n - (n - pN(0))(1 + p)^T)/p = 0
From here we get:
n = pN(0)/(1 - (1 + p)^(-T))
In our case, the given interest p' is annual. For it we have:
p = (1 + p')^(1/12) - 1
So:
n = ((1 + p')^(1/12) - 1)N(0)/(1 - (1 + p')^(-T/12))