Find the periodic payment R required to amortize a loan of P dollars over t years with interest charged at the rate of r%/year compounded m times a year. P = 16,000, r = 4, t = 6, m = 12

Question

Find the periodic payment R required to amortize a loan of P dollars over t years with interest charged at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.)

P = 16,000, r = 4, t = 6, m = 12

I have about 10 questions that are like this

if I could see an example of how to work this

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Answer 1

Let us denote the loan balance after (n-1) months L_n-1. During one month the interest of (r/m)% is added (m/12) times and after that monthly payment R is subtracted. Thus the loan balance after n months is

L_n = L_n-1[1 + (r/m)]^(m/12) - R.

Using geometric series formula we get

L_n = [Pq^n-1(q - 1) - R(qⁿ - 1)]/(q - 1).

Here q = [1 + (r/m)]^(m/12). The loan is payed off in n = 12t months if L_n = 0. Hence

R = Pq^{(n - 1)}(q - 1)/(qⁿ - 1) = 16,000_*(1 + 0.04/12)⁷¹_* (0.04/12)/[(1 + 0.04/12)⁷² - 1] = 249.47.