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?