interest rate is 12% per year compounded monthly = 1% per month.
you make the minimum payment of 5% of the balance at the end of each month.
your formula is that, at the end of each month, your remaining balance at the end of the current month is the remaining balance at the end of the previous month * 1.01 minus .05 * the remaining balance at the end of the previous month.
this formula becomes An = A1 * (1 + .01 - .05) ^ (n - 1) which becomes:
A2 = A1 * .96 ^ (n - 1)
this is a geometric series formula.
when A1 = 4000 and A2 = 0, this formula becomes:
0 = 4000 * .96 ^ (n - 1)
you will try to solve this formula in the following way.
divide both sides of this formula by 4000 to get:
0 = .96 ^ (n - 1)
take the log of both sides of this formula to get:
log(0) = log(.96^(n-1))
since log(.96^(n-1)) is the same as (n-1) * log(.96), this equation becomes:
log(0) = (n-1) * log(.96)
solve for (n-1) to get:
(n-1) = log(0)/log(.96)
since log(0) is not valid, you can't solve for n.
the reason for this is that you have an infinite geometric series.
An will approach 0 as n gets larger, but it will never be equal to 0.
therefore n approaches infinity as An approaches 0.
the best you can do is get it down to something very small, but not 0.
i chose to get it down to 1 penny.
the formula of An = A1 * (.96) ^ (n-1) becomes:
.01 = 4000 * (.96) ^ (n-1)
divide both sides of this formula by 4000 to get:
.01/4000 = .96^(n-1)
solve for (n-1) to get:
(n-1) = log(.01/4000) / log(.96) = 315.987006
i confirmed with excel that this is accurate.
the balance will be reduced to .01 in 315.987006 months.
here's what the excel printout looks like.
you can see that, at the end of the 15th month, the remaining balance is greater than a penny and at the end of the 16th month, the remaining balance is less than a penny.
if you graph the equation, you will see the following.
in the graph, x represents n.
eom is equal to n-1, therefore, when n = 316....., eom is equal to 315.....
you can see that in the table.