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?

• $1,2,4,8,16,32,\underline{~~}$ I think you mean instead $f(n)=2^n$, not $n^2$ (which follows $1,4,9,16,25,36,\dots$). The unfortunate answers to your questions are 1) Yes, 42 is always an acceptable answer for example, 2) Yes, every number is an acceptable answer (unless specified ahead of time that the available options are restricted to be from a finite set), 3) that's a tough one... I'll have to say no or at least, not easily for many cases. Recursive definitions are generally kinder to use than closed forms in several settings. 4) Far too subjective, I'd say you can't. – JMoravitz Apr 2 '17 at 1:58
• Oh, you've opened a can of worms. Mathematicians HATE these questions. It is ALWAYS the case that ANY number is a correct answer. So although the person asking thinks the next number is 1,2,4,8,16,32,64,, the next answer being 1,2,4,8,16,32, $\sqrt {\pi}$ is every bit as correct. So the answer to your questions 1) yes, there must always be EVERY number as an answer 2) yes, there is always an infinite number because every number is an answer. 3) yes, there are an infinite number of equations yielding different solutions, 4) you can't! It's a STUPID question. – fleablood Apr 2 '17 at 2:01
• By the way, the "solution" isn't $y=x^2$ it's $y=2^x$. – fleablood Apr 2 '17 at 2:02
• The only caveat for what's already been said is that if you say ahead of time what type of sequence you're dealing with, then yes there is such a thing as a 'correct' answer. For example, an arithmetic sequence has a constant difference between terms, so If I gave you the sequence $1, 3, 5, 7,\dotsc$ and specified that it was arithmetic, then the next term would need to be $9$. If that wasn't specified, then all bets are off and anything works. – DMcMor Apr 2 '17 at 2:05
• @fleablood I strongly disagree! :^) Some mathematicians have worked on this problem (e.g. Simon Plouffe here). For the OP: You should take a look at the Online Encyclopedia of Integer Sequence as well as the GuessGF function in maple (that guesses the generating function of a finite sequence). – Olivier Apr 2 '17 at 2:06

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.

• Well.... time to change my password again. – JMoravitz Apr 2 '17 at 2:29