# simple math pattern

• Sep 19th 2006, 04:12 PM
Devin Brugger
simple math pattern
• Sep 19th 2006, 04:30 PM
ThePerfectHacker
Quote:

Originally Posted by Devin Brugger

a_n=-(1/2)x^4+(11/2)x^3-(31/2)x^2+(41/2)x-10
So,
a_0=0
a_1=5
a_2=20
a_3=48
a_4=80
---
It follows the rule of a quartic plynomial.
• Sep 19th 2006, 05:14 PM
topsquark
Quote:

Originally Posted by Devin Brugger

I can't find a pattern for 0, 5, 20,... but I can find one for 0, 6, 20, 48, 80

Typo?

-Dan
• Sep 19th 2006, 05:18 PM
dan
ok PH, i'm impressed....how in the world did you come up with that?? what is the strategy to finding patterns..??
• Sep 19th 2006, 05:34 PM
ThePerfectHacker
Quote:

Originally Posted by dan
ok PH, i'm impressed....how in the world did you come up with that?? what is the strategy to finding patterns..??

Thank you, I am humanities gift from the gods I cannot tell you how I found that.

Okay, okay I be honest.

I used something called "method of least squares" for polynomials. It works for other class of functions. When I use this I looks whether or not this function is "good" looking. Meaning are the coefficients nice or ugly? If nice like here I assume that is the pattern. CaptainBlank referrs to it as the "Lagrange Interpolating polynomial".

In fact I can prove that given a sequence:
a_0,a_1,...,a_(n-1)
We can find a unique polynomial having a degree up to (n-1) with precisely satisfies this equation. (The prove is based on being able to solve a system of linear equations).
• Sep 20th 2006, 12:36 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
a_n=-(1/2)x^4+(11/2)x^3-(31/2)x^2+(41/2)x-10
So,
a_0=0
a_1=5
a_2=20
a_3=48
a_4=80
---
It follows the rule of a quartic plynomial.

If you start from 0, then the first element in the sequence would be -10
from your quartic. You mean:

a_1=0
a_2=5
a_3=20
a_4=48
a_5=80

RonL
• Sep 20th 2006, 12:42 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
Thank you, I am humanities gift from the gods I cannot tell you how I found that.

Okay, okay I be honest.

I used something called "method of least squares" for polynomials. It works for other class of functions. When I use this I looks whether or not this function is "good" looking. Meaning are the coefficients nice or ugly? If nice like here I assume that is the pattern. CaptainBlank referrs to it as the "Lagrange Interpolating polynomial".

No I don't, the Lagange interpolating polynomial is constructed to exactly
go through the points. It just so happens that a least squares polynomial
fit of a degree n-1 polynomial to a squence of n points must also go through
the points and so in practice is identical to the Lagrange polynomial, but
they differ in their method of construction.

Here the least square and the Lagrange polynomials are the same but the
distinction between them remains as it lies in their methods of construction.

RonL

RonL
• Sep 20th 2006, 04:29 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
a_n=-(1/2)x^4+(11/2)x^3-(31/2)x^2+(41/2)x-10
So,
a_0=0
a_1=5
a_2=20
a_3=48
a_4=80
---
It follows the rule of a quartic plynomial.

The problem is that this sort of solution is almost never the solution to this kind of problem. Odds are there is some method to construct the terms of the series and your polynomial will not predict the next one. (I tried this once as the solution to a "Math Challenge" and the professor's granted that I had found a solution, just not the right one. :) ) Though I will grant it was a nice piece of work. ;)

-Dan
• Sep 20th 2006, 05:44 AM
CaptainBlack
Quote:

Originally Posted by topsquark
The problem is that this sort of solution is almost never the solution to this kind of problem. Odds are there is some method to construct the terms of the series and your polynomial will not predict the next one. (I tried this once as the solution to a "Math Challenge" and the professor's granted that I had found a solution, just not the right one. :) )

Which is why I usually respond what is the next term questions with what
would you like it to be.

RonL
• Sep 20th 2006, 06:05 AM
topsquark
Quote:

Originally Posted by CaptainBlack
Which is why I usually respond what is the next term questions with what
would you like it to be.

RonL

That's two double posts in one day for you, Captain. Is your evil twin on the computer again? :)

-Dan
• Sep 20th 2006, 06:10 AM
CaptainBlack
Quote:

Originally Posted by topsquark
That's two double posts in one day for you, Captain. Is your evil twin on the computer again? :)

-Dan

It's the sluggishness of the world wide wait (at least at work in the
lunch hour:mad: ) results in me editing a post while the original is uploading
without noticing, then hitting submit again:o

Its also a type of double posting I don't mind too much when others do
it. Near identical posts in the same thread (of forum) are usually finger
trouble, best way to deal with them is to just delete the duplicate.

RonL