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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?