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Talking about How to solve the sequence: $87, 89, 95, 107, ?, 157$ for example, I read the hint:

The difference between each term goes like this: 2,6,12. Can you notice any pattern? Based on it, can you prove that the next term in it is 20?

and then I could solve it.

My question is how to solve these type of questions without guess work.

What topic of Maths or thinking process would enable me to understand and solve these type of problems?

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Aquarius_Girl
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    In general, the whole point of these sorts of questions is to guess a pattern that you think is likely to be the solution. In actuality, there are many 'patterns' or 'rules' that fit any particular sequence. – Will R Aug 01 '15 at 09:25
  • @WillR Would you mind listing them so that I can get a start point at least? – Aquarius_Girl Aug 01 '15 at 09:26
  • I believe some have been listed in the top-voted response to the question you linked to. – Will R Aug 01 '15 at 09:34
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    If you want to get good at solving these kinds of problems, the best way to go about it is to just keep trying; do lots of problems and you will get the hang of it eventually. Tricks like 'looking at the sequence of differences', like that used above, or 'looking at the sequence of differences of differences' can help, but you can't expect those strategies to be reliable - that's my point. – Will R Aug 01 '15 at 09:37
  • Intriguingly, "most" sequences containing 87,89,95,107 have 152 where the problem statement expects 157 – Hagen von Eitzen Aug 01 '15 at 10:29
  • It's a bit like with integration problems, or certain kinds of number theory problems: yes, there are plenty of examples where we can find the answer through some method/technique/way of thinking, but, in general, we can't always find a solution, and sometimes there is no solution (and sometimes there is a 'solution', but only because we say so)! Nevertheless, it is possible to get very, very good at solving such problems in practice simply through experience. – Will R Aug 01 '15 at 11:47

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You can try first to check the difference between the terms if it is constant then it is done. Sometimes the difference has a relation also for example take the sequence $4, 7, 12, 19, ? , 39$ the differences are $3, 5, 7, \cdots$ which are the odd numbers hence you can guess it is $? = 19 + 9 $.

Secondly you can think about dividing the terms it will something similar for the above sometimes it is a constant and sometimes you can find a relation.

IrbidMath
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