I agree with Vivian on the expected answer to this problem.
However, I have a pet peeve with this kind of problem "predict the next 3 numbers". When I first encountered this kind of problem in school, I kept getting the nagging feeling that I was missing something; the question didn't seem to make any sense. I could, I thought, pick any next numbers. Why did the numbers need to follow a formula? It turned out my intuition was correct, but I couldn't get away with saying so until I learned more mathematics.
In the problem 4 numbers are given: 3/1, -9/4, 27/16 and -81/64. Most textbooks will tell you that this is the first part of a geometric series with the first term 3 and the ratio (-3/4).
But if we start taking differences, there are three first differences -21/4 , 63/16, -189/64. There are two second differences, and only one third difference. Because there is a single third difference, and let's take an open mind about the rest of the series for a minute, the assumption that all third differences are the same is consistent with the terms of the series that we see.
Now we know that if a function has a constant third derivative, then it is a third degree polynomial. So there is a polynomial of degree 3 p(x) which has the property that p(1) = 3, p(2) = -9/4, p(3)= 27/16, and p(4) = -81/64, and that polynomial will not follow the geometric series because it goes to -infinity at one end and to + infinity on the other. And I can use that polynomial to give me the next three values p(5), p(6) and p(7) which will all differ from the geometric series.
In fact, there are a number of series that match the four given numbers, not just polynomials.
The truth is, the question is a bogus question. The real question being asked is "guess the series that the question writer has in mind", and if you are a smart cookie you will guess that the teacher is probably thinking of the geometric series and not the polynomial and not other solutions. In the end, this is a subjective problem, not an objective one and so on some level I think it's an unfair question.
