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$3, 4, 10, 33, 136$

what will be next most appropriate value? I tried finding any relation in the sequence but i couldn't.

$a.276 $

$b.539 $

$c.612 $

$d.685$

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    I agree that questions like this can be frustrating, especially when multiple answers might fit, but I don't agree that they're completely worthless. I think the ability to look at arbitrary data, see patterns, find possible relationships, and then identify the most likely relationship is a critical skill for a mathematician to have. The question isn't "what was the exam writer thinking when he wrote this?", it's "what's the simplest relationship you can find between these numbers?"2011-05-03

2 Answers 2

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I agree with all the complaints about this sort of problem, but still.... There are some techniques which work from time to time.

Try taking differences: $4-3=1$, $10-4=6$, $33-10=23$, $136-33=103$, so now we have to explain the sequence $1,6,23,103,\dots$. Hmm, that doesn't seem very helpful.

OK, subtraction didn't work, try division: $4\div3=1r1$, $10\div4=2r2$, $33\div10=3r3$, $136\div33=4r4$ - hey, that looks much better! (When I write $arb$, I mean quotient $a$, remainder $b$.)

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    Wow! I never knew division could be applied. I always used subtraction. +1 for *arb* notation. Its really helpful.2011-12-21
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I must say, I have always disliked 'find the next term in the series question'. For any sequence, it is easy to produce any number next (e.g. for a sequence of $n$ terms, pick the $n+1$ number and then fit a polynomial to those $n+1$ terms).

OEIS does not give anything useful for your sequence - how has it arisen?

Edit: For this question, as Moron has shown, the likely answer is 685, based on the sequence $3,3\times 1 + 1 = 4, 4 \times 2 + 2 = 10, 10 \times 3 + 3 = 33, 33 \times 4 + 4 = 136,$$136 \times 5 + 5 = 685$ . But in general knowing how to find the pattern in this sequence, will not help (much) in finding patterns in similar sequences.