A walk down memory lane

I just found a page on "How to Find a Formula for a Set of Numbers".  It's a cool little procedure for taking a series, like:

2, 8, 9, 11, 20

and producing a polynomial to give you the next ones in the series, like:

n³- 17/2 n²+ 49/2 n - 15

where n is the term number, starting from n=1.  Try it out!  Anyway, it was a method I learned in high school math league, and thought it was so cool I wrote a BASIC program on the old TRS-80 computers to do it.  I had forgotten how to do it, and it was fun to see it again.  I particularly liked the comment on the page:

"""If someone gives you the sequence, say, "1, 4, 9, 16", you could run them through the above process and get the answer that the person is probably looking for: the rule is n² so the next value is 25. But you could also invent any number as the next number in the sequence, say 42, and come up with a rule for "1, 4, 9, 16, 42". Feel free to work it out. It comes out to:

 

17/24 n⁴ - 85/12 n³ + 619/24 n² - 425/12 n + 17

and the next term is then 121.

So if you want to be obnoxious, the next time you are given a quiz of "find the next number in the series" problems, just pick any number you like and fill it in, and you'll be completely correct. You'll probably get a failing grade on the test, but you can enjoy the smug satisfaction of knowing you were right."""

I knew a kid who, because of a ridiculous fluke, had to redo some of his middle-school competency tests in high school.  So, when presented with a series like 2,4,6,8,... he did this on a test (and yes he did fail the test and have to redo it).  He was also shown a number of clocks, and asked what time does this show, and for all of the answers put "analog time".