How to find nth term of the sequence 6+13+24+39+.....
For your problem without any more clarification there will be an infinite number of sequences that fit your description.
You could find one by fitting a polynomial through them or use a program to generate a sequence.
Without any more constraints, a unique sequence is not really possible.
I would do this using "Newton's divided difference formula". The numbers are , , , and . The "First Differences" are , , and then the "Second Differences" are , . If we assume that the sequence continues so that the second differences are alway four (and the third and higher differences are 0) we can write, in analogy with the Taylor's series, .
You will note that when n= 0, , when n= 1, , when n= 2, , and when n= 3, exactly as we want.
You could also use the fact that, since the second differences are constant, the sequence is a quadratic function of n: . Then use ,
and to solve for a, b, and c.