Let's use P as the principal amount of $30000.
Before the first instalment is made, a month's interest will have been added. The annual rate is 6% so the monthly rate is 6/12=0.5%=1/20=0.005. The principal plus interest is P+0.005P=1.005P. Let's call the monthly payment M.
We'll call the new principal after one monthly instalment has been paid P1: P1=1.005P-M. This is smaller than the original so the interest is gradually decreasing with each month. The interest rate doesn't change, but the amount of interest slowly decreases.
So, when the second instalment is paid the principal changes again: P2=1.005P1-M.
So P2=1.005(1.005P-M)-M=1.005^2P-1.005M-M
P3=1.005P2-M=1.005(1.005^2P-1.005M-M)-M=1.005^3P-1.005^2M-1.005M-M.
Now we see a pattern.
For some new principal Pn the original principal has grown to (1.005^n)P where n is the number of months.
And the monthly payment has turned into a series: M(1+1.005+1.005^2+1.005^3+...+1.005^(n-1)) to be subtracted from (1.005^n)P. The series for M can be summed=M(1.005^n-1)/(1.005-1).
In this problem n=48 months (4 years). So the series for M=M(1.005^48-1)/0.005=54.098M.
Since the loan must be paid off in 4 years Pn=0, so 54.098M=1.005^48 * 30000=38114.67.
We can now work out M=38114.67/54.098=$704.55.
The monthly instalment is therefore $704.55.
