Just think it through, one payment at a time.
From the start, I'm not absolutely certain there was a payment ON the 18th birthday. let's assume that there was. If we get $500 too much, we'll discard this payment.
Starting from the last payment and working backwards, we have:
18th: $500 and no accumulation for interest
17th: $500(1+i) -- 1 year's accumulation for interest
16th: $500(1+i)^2 -- 2 year's accumulation for interest
15th: $500(1+i)^3 -- 3 year's accumulation for interest
Birth: $500(1+i)^18 -- 18 year's accumulation for interest
With any luck, one should notice this is a Geometric Sequence and we should be able to add them all up.
$500 + $500(1+i) + $500(1+i)^2 + $500(1+i)^3 + ... + $500(1+i)^18 =
$500(1 + (1+i) + (1+i)^2 + (1+i)^3 + ... + (1+i)^18) =
Our only remaining concern is 'i'. What is it? The formula above uses 'i' as an annual effective interest rate. We need to find one of those. We are given 8% Nominal Interest and Monthly Compounding. This gives:
As can be seen, $21,380.97 - $500.00 = $20,880.97, so I guess there was not a payment made ON the 18th Birthday Anniversary. This leaves us with a bit of a dilemma. On a written exam or a homework assignment, I would state my assumptions and provide both answers, citing the ambiguity of the word "until". Anything marked wrong would get a vigorous challenge. On a multiple-choice exam, I would be prepared to find either answer. In my view, if both appear on the multiple-choice exam, the question probably should be discarded as accepting either answer will not necessarily provide any information about a student's knowledge. One may simply have done it badly. In any case, questions should be clear. If you have discussed the word "until" in class, and it has been defined clearly to mean "NOT on the end date", then you can be expected to get the unique value. There may also be a diagram explaining the intent. It is a very hard thing to write perfectly clear questions. It is up to the student to explain any point of ambiguity. The exam writer cannot be expected to think of every possible translation, but I'm sure the exam writer tries to do that.
Well, enough of exam philosophy...