# what is the nature of this sequence?

$3, 4, 10, 33, 136$

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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Is this just a puzzle or do you have some context you can provide? – Aryabhata May 3 '11 at 6:15
I have got options with me. $a.276$ $b.539$ $c.612$ $d.685$ – amul28 May 3 '11 at 6:18
One option is $685$. $3, 3*1+1 = 4, 4*2 + 2 = 10, 10*3+3 = 33, 33*4+4 = 136, 136*5 + 5 = 685$... But such questions are nonsense as Qwirk's answer shows. I am voting to close a NARQ. – Aryabhata May 3 '11 at 6:21
@amul: I really don't know. There is no systematic method. Really, such questions are nonsense and solving this one likely won't help you solve other nonsense questions like this. It is unfortunate, but you have to guess what the idiot who wrote the question on the entrance test was thinking of... – Aryabhata May 3 '11 at 6:41
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?" – Kevin May 3 '11 at 15:15

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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seems very clear for me. – amul28 May 3 '11 at 7:09
Wow! I never knew division could be applied. I always used subtraction. +1 for arb notation. Its really helpful. – Shiplu Dec 21 '11 at 7:44

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.

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