Math Is Fun Forum

bossk171

Member

Registered: 2007-07-16

Posts: 305

2,4,8, what...

The first three numbers in a sequence are:

2,4,8,

What is the next one after that? How many different answers can we come up with?

Rule #1: For the sake of entertainment post the answer in as the "face" of a hide tag, and keep how you got it hidden. For example:

Rule #2: Any sequence you find/create must be evident in the first three terms.

Ex: the doubling sequence is ok because 2 doubled is 4 and four doubled is 8
Ex: f(x) = f(x-1) + f(x-3) is NOT ok because it assumes that all of the given terms of a sequence are the starting terms.

You're allowed to use two of the given terms as starting terms in a sequence (kind of like the Fibonacci sequence requires two starting terms) but all three cannot (otherwise things get boring).

I've come up with a few answers, but will wait to post them. All the answers I can come up with are even, can anyone come up with an odd one?

My math jargon isn't very good, so if anyone thinks the wording should be edited, please tell me... game on!

Last edited by bossk171 (2008-04-07 04:31:23)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: 2,4,8, what...

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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..."

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: 2,4,8, what...

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Wrap it in bacon

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: 2,4,8, what...

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

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: 2,4,8, what...

Hooray to me for being the first to post a non-increasing sequence!

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: 2,4,8, what...

I propose a ban on interpolation.

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

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: 2,4,8, what...

I agree mathsy, but give Jane a little credit where Lagrange is due.

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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..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: 2,4,8, what...

I believe that Jane gets bonus points for a cool answer, but I think it's kind of cheating. Can Ricky or Mathsyperson please please offer up a good definition of interpolation, one which people with lower math skills can understand?

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: 2,4,8, what...

Lagrange’s interpolating method allows you, when given a sequence of k numbers, to find the unique polynomial L(x) of degree at most k−1 such that L(1), L(2), …, L(k) are precisely the numbers of the sequence taken in order.

When k is small, it’s easy to compute the Lagrange polynomial by hand. Take your example, with k = 3. What’s a polynomial L(x) of degree at most quadratic such that L(1) = 2, L(2) = 4, L(3) = 8? Well, that’s obvious. It’s

which simplifies to

An alternative method of computing the Lagrange polynomial is to let

and solve for the a, b, c by substituting values of x. You will end up with exactly the same polynomial as above. The Lagrange polynomial is unique up to quadratic degree.

Hence a simple polynomial formula for your sequence is

If you just take the Lagrange polynomial, the next number in your sequence will be

. But here comes the trick. By adding an extra factor and making the polynomial cubic (i.e. raising the degree by 1), you can literally choose any number you like as the next number in the sequence! Yes, literally!

So let’s say you want the next number of your sequence to be 0. Since the fourth number would be 14 if you simply used the Lagrange polynomial, you want to the extra factor to be −14 when n = 4. Also, this extra factor must be 0 when n = 1, 2, 3. Thus, it’s obvious what your extra factor must be:

Now this formula will give you 2, 4, 8, 0 as the first four terms of your sequence. What’s more, to impress your friends, you can simplify the formula to

removing all traces of cheating and causing your friends to think you’re a genius.

When the number of terms of your sequence get bigger, the calculations get much more complicated, but it’s still nothing that a good program can’t handle. So get yourself a good computer program for this if you really want to impress your friends.

Last edited by JaneFairfax (2008-04-07 23:51:20)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: 2,4,8, what...

Hey hey, I got another one!

Complex numbers!

AND without interpolation!

Last edited by JaneFairfax (2008-04-07 22:25:37)

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: 2,4,8, what...

Wow, this interpolation business is amazing! Along with Langrange Multipliers, that's two brilliant things out of two I've learnt from this guy.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: 2,4,8, what...

Cool. I knew about

but had cheated and used Excel to do the grunt work for me. I guess no more interpolation then.

Last edited by bossk171 (2008-09-19 07:40:02)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: 2,4,8, what...

edit: oh got another one

Last edited by Kurre (2008-04-09 06:11:46)

smellyellie123

Member

Registered: 2006-11-08

Posts: 7

Re: 2,4,8, what...

you could do 2,4,8,16,  or  2,4,8,12  or  2,4,8,14!

i have probably already been beaten!!!!!!!!!!!!!!!!!:D

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: 2,4,8, what...

to bump this thread, here is another sequence:

Last edited by Kurre (2008-04-15 03:14:28)