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How can I write an equation that matches any sequence?

I'm learning sequences right about now, and I'm having a really hard time finding the formula for a given sequence of numbers. I'm worried that, at exam time I'll be spending as much time finding the formula as I am now.

For example, one problem gave me:

$$\langle a_n \rangle = \\{2, 8, 18, 32, 50\\}$$

Okay, it's giving me a Math Processing error. I'm sure the code is correct; I'm also seeing this error on other people's pages. If it's me, please let me know.

Again, one problem gave me the sequence

(an) = { 2, 8, 18, 32, 50 }

I spent a good ten to fifteen minutes tinkering around, making some diagrams, plugging in the values. I'll post some examples:

a1 = $2$ : $+6$ : $+4$

a2 = $8$ : $+10$ : $+4$

a3 = $18$ : $+14$ : $+4$

a4 = $32$ : $+18$ : $+4$

a5 = $50$

I first checked to see by how much each values differs to the next. The +4 was the increment by how much more it increased every time.

I therefore came up with the following solution (among many others):

$2a_n + 4a_{n-1}$

This worked for the first three values in the sequence. Then it broke. Damn.

Eventually I gave up and looked at the solution:

$a_n = 2n^2$

Seriously? I would never have guessed this. So my question is:

How are you supposed to find a pattern? Are there any good methods for doing so?

Simon Fraser
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  • It looks like you are on your way to discovering finite differences. That and other techniques are described in answers to this question. If you don't find those answers helpful, please add more details to your question to distinguish it from the duplicate. – Larry Wang Sep 01 '10 at 15:15
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    Knowing the answer of $(a_n)={2,8,18,32,50}$, we could divide by 2 to get $(a_n/2)={1,4,9,16,25}$ and now the pattern is clear. My opinion is that for this kind of problems one can never be assured if there is only one correct formula, since that depends on the elementary functions you use to get the formula. – Américo Tavares Sep 01 '10 at 15:20
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    To spoil the fun, http://oeis.org/search?q=2,8,18,32,50 – kennytm Sep 01 '10 at 17:40

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