Use the formula for the present value of an ordinary annuity

Solution

Present Value, P = 5000
Payment (monthly) A = 500
interest (monthly) i = 0.035
The amortization formula would equate future value with the sum of all the payments, all increased at rate of interest i.
Future value = sum of all payments
Let R=1+i = 1.035
PRⁿ = A + AR + AR² + AR³ + ... + AR^n-1
=A(Rⁿ - 1)/(R - 1) (by factoring)
Hence
(Rⁿ-1)/((R-1)*Rⁿ) = P/A
To solve for the period n, there is no explicit formula to calculate.
The easiest way is to calculate the payment for a given period n.
If the payment matches 500, then the estimated n is correct.
For example,
The equation can be converted into a formula for the monthly payment, A
A=P(R-1)R^n/(R^n-1)
For
P=5000
R=1.035
we make a first estimate from
5000/500 = 10
We know n > 10, so try 13
A=5000(0.035)1.035^13/(1.035^13-1)
=485.3 < 500
So we try 12 payments
A=517.4
We then know that the period n lies between 12 and 13, and for all practical purposes, we would put it at 13.