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This is a general question which I'm not really quite sure is appropriate here so before you decide to down vote, just drop me a message that it's not appropriate and it will be promptly removed or relocated.

Now on the question:
Are there any programs that can crack most number sequences for the purposes of finding the answer? I often purchase second hand puzzle books, and they often don't have answers in them or the rule for any given nth term. The problem is when I encounter a sequence I can't solve, I don't have a solution to work off or a AP/GP rule to work off of. So the focus of the question is are there any programs that can be given an input and give possible answers and possibly a rule, although the rule is unnecessary, as I can work that out myself.

Many thanks in advance.

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    $\begingroup$ I think OEIS is a great source... $\endgroup$
    – ABcDexter
    Commented Jun 18, 2016 at 5:18
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    $\begingroup$ I personally like your question and asking about puzzle-making or solving as a process is absolutely okay on site. However, asking for what are best resources of XY are generally not appreciated on StackExchange (opinion based) so, I'm curious how this question will fare on Puzzling... You might consider switching over to the meta-site though. There is a post about puzzle solving sources. $\endgroup$
    – BmyGuest
    Commented Jun 18, 2016 at 6:13
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    $\begingroup$ any partial sequence of $n$ terms can be fitted to a polynomial or order $n-1$ $\endgroup$
    – Jasen
    Commented Jun 18, 2016 at 6:36
  • $\begingroup$ @Jasen You are absolutely right (it's called Extrapolation). $\endgroup$
    – ABcDexter
    Commented Jun 18, 2016 at 6:42
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    $\begingroup$ In a general case, Douglas Hofstatder wrote about a program he was working on called seek-whence to do this. $\endgroup$
    – Ross Millikan
    Commented Jun 18, 2016 at 15:21

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I think it all comes with practice. You might solve a thousand number sequences, think you're good and then encounter a new one which might baffle you. But the hard-work that you've spent in solving those thousand will get rewarded, eventually.

There are some good number sequence problems which do not come under standard AP,GP,HP or AGP or any other well known mathematical sequence/pattern. Also, there are some stupid so-called "sequences" which are meant to be not solved but use that lateral-thinking part of our brains.

And to answer your question, there are some really good sources like:

  1. OEIS(The On-Line Encyclopedia of Integer Sequences) which has been my good friend for four years.
  2. Wolframalpha, you can check it as it's in its beta.
  3. github/nextTerm...

(Do add more, but really these are good enough :) ).

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