Any sequence of numbers is a valid sequence; so you could put any number you want next and this would be a valid sequence.
In some such problems, there is enough evidence with the given numbers to figure out what is LIKELY TO BE the next number; but there is no way to know for sure.
Any finite sequence of n numbers can be produced by a polynomial function of degree (n-1). 2 numbers give you a polynomial of degree 1 (think of it as 2 points determining a line). 3 numbers can be produced by a quadratic polynomial (degree 2). 4 numbers can be produced by a cubic polynomial, 5 numbers by a polynomial of degree 4, and -- as in your example -- 6 numbers by a polynomial of degree 5.
So you could take the general polynomial of degree 5...
... and form 6 equations by setting P(1)=5, P(2)=10, ..., and P(6)=61. You can then solve that system of 6 equations in 6 unknowns to determine the particular polynomial that produces your sequence.
That method will always give you "A" solution for the sequence. But there is never any way to know if that is the "right" answer according to the author of the problem.
Whenever I see a problem like this, I spend a very short time trying to see if I can see a nice pattern in the numbers; if not, I drop the problem, knowing I have better things to do with my time.
