Is there always an answer to "what is the next number in the sequence" This is my 1st post. I am not a mathematician- so please 'dumb down' any answers :-)
I having been thinking about these sorts of "IQ" puzzles:
What is the next number in the sequence: $1,2,4,8,16,32, \dots$
In this case, $y=2^x$
If I pick a series of (for example) $5$ random numbers and then ask you to find the 'next' number in the sequence...
$1)$ Will there ALWAYS be at least $1$ answer? e.g. $-111, 0,50,-112,77,1, \dots$
$2)$Is there perhaps always an infinite number of correct answers? e.g. $1,2,3,\dots$the answer could be 4 $(y=x+1)$ or $5$ if Fibonacci
$3)$ Is there always an 'equation' that will generate the 'answer' e.g. in the sequence $1,1,1,1,5,5,5,5,9,9,9,9,\dots $ I could state "the pattern is to repeat the number $4$ times, then add $4$" but can this be expressed as an equation rather than a 'program'?
$4)$ How to define an answer as 'correct'... e.g. mathematically simpler? 
 A: Such a question might get more traction over on puzzling stackexchange in terms of how to generally "solve" these sorts of things. As mentioned in some of the comments, mathematicians are generally reluctant to touch such things because really any number could be the next entry (unless we're working in a specific context of arithmetic / geometric / ... sequences). 
Mathematically, there is no provably correct answer---they best you can do is some judicious application of Occam's Razor. As Von Neumann once quipped, "With four parameters I can fit an elephant, and with five I can make him wiggle his trunk." We can always cook up a messier function that will work. 
It is also possible to view the solution as minimizing entropy---very roughly speaking, how much we learn with the next entry / how surprised you are. For example, consider the sequence $1,2,4,8,16,32,...$. How shocked would you be if I told you the next entry was ...


*

*$64$

*$48$

*$\$3.50$ 

*$\sqrt{\pi}$

*A picture of a platypus. 


As another example, if you found out that the first entries of my password were 1234passwer_, where you can't read my handwriting to make out what _ is. There is a clear guess that would be unsurprising. But it wouldn't be possible to prove with absolute certainty what that character has to be. 
