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$B=$ Balance after n payments

$L=$ Loan amount

$i=$ Interest rate per period

$P=$ Payment

The formula I found online at the website https://brownmath.com/bsci/loan.htm is:

$B=[L(i+1)^n]-\frac{p}{i}[(i+1)^{n}-1]$

I have several questions. The first question is does this formula work for all loans that are amortized. So amortized car loans, bank loans, student loans, and mortgages? When does this formula apply? I found several different formulas for mortgages and other stuff. Its almost like each type of loan has its own formula. The website called the above formula the "master formula".

The second thing is the website had a question:

You have $18,000 car loan at 14.25% for 36 months. You have just made your 24th payment of 617.39 and would like to find the payoff amount:

The answer to the question says the interest rate per year is $1.1875$ per month so $i=.011875$ But how is this the case?

I get $i= \frac{.1425}{36}= .0039583333$

Is there an error on the website or am I making a really stupid error in my calculation?

What happens to the formula when you have a down payment? Do you just subtract the down payment from the principal loan amount and use the formula to find the remaining balance after n payments?

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