That's such a cool idea. Do you then prove it is the formula you need, or do you just assume that if it matches for 10 terms it's probably the formula?

A math professor I had in college, Dave Kelly, had a trick where he'd show a seemingly reasonably-behaved sequence, explain where it came from, then show that the rest of the sequence was absurd -- like, it would be exponential for a while and then zero, or increase arithmetically for a while and then become undefined, etc. I wish I had taped that.