Math Is Fun Forum

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Sequences

Finished off these two pages:

Sequences and Series
Sequences - Finding The Rule

How do they look? Any mistakes?

(And, yes, not much on "Series" .. I hope to do a page or two on that in the future)

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Sequences

It should be noted in the 2nd one that even piece wise rules can be applied.  My favorite example of this is:

Find the next number in this sequence:

1, 4, 9, 25

Did you guess 36?  Because that's wrong.  The right answer is 73.  The rule in this sequence is:

x_n = n^2, unless n = 5 in which case x_n = 73.

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"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..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Sequences

...and unless n=4 in which case x_n = 25.

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Why did the vector cross the road?
It wanted to be normal.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Sequences

hehehe!

recursive definitions are also valid, right? like the fibbonacci series.

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A logarithm is just a misspelled algorithm.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Sequences

Recursive 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.

PS. To understand recursion, you must first understand recursion.

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Why did the vector cross the road?
It wanted to be normal.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Sequences

I should at least mention recursion, shouldn't I?

(But maybe I should mention it beforehand ... thanks mathsy!)

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Sequences

PS. To understand recursion, you must first understand recursion.

You always curse before you recurse.

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"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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Sequences

I don’t think the “finding the rule” section is relevant at all.

If you see a sequence beginning 1, 4, 9 … how on earth can you conclude that the next term must be 16? The rule might not be x[sub]n[/sub] =  n[sup]2[/sup] at all. It might be x[sub]n[/sub] =  (n[sup]3[/sup]+11n−6)⁄6 instead – in which case the next term is not 16 but 17.

In order to define a sequence, you need to define all its terms. You don’t just list the first few terms and ask everybody else to “find the rule”.
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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Sequences

In order to define a sequence, you need to define all its terms. You don’t just list the first few terms and ask everybody else to “find the rule”.

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.

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"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..."

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Sequences

In order to define a sequence, you need to define all its terms. You don’t just list the first few terms and ask everybody else to “find the rule”.

But then we may as well throw the question away (and with it most IQ tests!)

And people do get asked "find the next number in the sequence", and in that case I think the simplest rule would provide the best answer (but mention that there are others).

And when we do get visitors asking questions like that it would be nice to have a page to refer them to ... so if anyone has any neat tricks to include, let me know.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Sequences

The first rule you give:

"Find the rule of 1, 2, 4, 8, 16, 32, 64"

Isn't that

, not

?

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Sequences

Yes, you are right! Thanks Identity.

I rewrote the example and made it 1,4,9,16, ...

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman