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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):
2an + 4an-1
This worked for the first three values in the sequence. Then it broke. Damn.
Eventually I gave up and looked at the solution:
(an) = (2n2)
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?