Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20070903 08:39:42
SequencesFinished off these two pages: "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #2 20070904 01:37:10
Re: SequencesIt should be noted in the 2nd one that even piece wise rules can be applied. My favorite example of this is: "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #3 20070904 01:58:52
Re: Sequences...and unless n=4 in which case x_n = 25. Why did the vector cross the road? It wanted to be normal. #4 20070904 02:02:41
Re: Sequenceshehehe! A logarithm is just a misspelled algorithm. #5 20070904 04:19:01
Re: SequencesRecursive definitions are kind of what MathsIsFun is calling 'find the rule'. As he says in the page, they're valid, but an nth term is more useful. The Fibonacci sequence does have an nth term, but it's considerably more complicated than its recursive one. Why did the vector cross the road? It wanted to be normal. #6 20070904 07:17:20
Re: SequencesI should at least mention recursion, shouldn't I? "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #7 20070904 07:44:13
Re: Sequences
You always curse before you recurse. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #8 20070904 07:58:28
Re: SequencesI don’t think the “finding the rule” section is relevant at all. #9 20070904 08:45:49
Re: Sequences
Technically, you're right. But I think you're missing the bigger picture. In many combinatorial problems which have solutions for n=1, 2, 3..., what you typically do is list out the first few terms, find the rule, then prove that the rule applies. Being able to "see" the rule only after the first few terms is a very important skill and one that needs to be practiced. In general, many problems in pure mathematics also work like this where you can prove it for a few simpler cases and then notice recurring themes to do the entire proof in general. This is especially true with algorithms in graph theory. Not exactly the same thing, but a close parallel. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #10 20070904 09:21:36
Re: Sequences
But then we may as well throw the question away (and with it most IQ tests!) "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #11 20070904 17:27:10
Re: SequencesThe first rule you give: #12 20070904 21:23:59
Re: SequencesYes, you are right! Thanks Identity. "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman 