Formulas for Series: Connect Patterns with Math Equation

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Discussion Overview

The discussion revolves around finding methods or formulas to connect numerical patterns through mathematical equations. Participants explore various approaches to derive formulas for sequences, specifically focusing on the example sequence 1, 2, 4, 7, 11. The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant requests methods or formulas to connect patterns, indicating a need for simple examples.
  • Several participants express that the initial question is too vague and seek clarification on the specific requirements.
  • A participant suggests a piecewise function to define the sequence values explicitly for given inputs.
  • Another participant discusses the concept of "first differences" and "second differences," noting that the second derivative is constant, hinting at a specific type of function.
  • One participant mentions "Newton's Divided Difference" interpolation formula as a potential method to find a polynomial for any finite sequence of values.
  • A later reply emphasizes that there is no single method to determine the general term of a sequence from a finite number of terms, highlighting the existence of multiple sequences that can fit the same values.
  • Another participant suggests that if a simple polynomial formula is sufficient, Lagrange's formula or Newton's divided difference formula could be applicable.

Areas of Agreement / Disagreement

Participants generally express disagreement on the existence of a single method for determining a sequence's general term, with some proposing specific formulas while others highlight the complexity and variability of potential solutions.

Contextual Notes

Limitations include the ambiguity of the initial question and the dependence on the definitions of terms and methods discussed. The discussion does not resolve the mathematical steps or assumptions involved in deriving the formulas.

mubashirmansoor
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I'd be glad if someone would provide me the ways or formulaes to connect a certain type of pattern through a mathmatical equation.

The simple ones are enough too.

Thankyou.
 
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The question is way too general, what are you looking for ?
 
Your question is far too vague to be answerable.
Besides, use either formulae or formulas.
 
What I mean is actually any method or a formulae for conecting a pattern, as an example ; 1,2,4,7,11... any method to connect these numbers by a certain formulae?
 
mubashirmansoor said:
What I mean is actually any method or a formulae for conecting a pattern, as an example ; 1,2,4,7,11... any method to connect these numbers by a certain formulae?
Try this:

Code: 
f(x) = 1  if  x = 1
       2  if  x = 2
       4  if  x = 3
       7  if  x = 4
       11 if  x = 5

Other than that, I think you are looking for this website:

http://www.research.att.com/~njas/sequences/
 
Last edited by a moderator:
You can define it recusively as

[tex]\left\{\begin{array}{l}a_0=1\\a_n=n_{n-1}+n\end{array}\right.[/tex]
 
Last edited:
Thankyou dav but I'm looking for an overall method
 
Well, the "first differences" are 1,2,3,4, ... The "second differences" are 1,1,1,1, ... In other words, the second derivative is constant.

So, what kind of function has a constant second derivative?

Once you get this, it's pretty quick for any sequence of this type. I'll let you figure it out, since this smells like homework.
 
The only thing I can think of that is close to what you appear to want is "Newton's Divided Difference" interpolation formula. It can be used to find a polynomial that will give any finite sequence of values for n= 0, 1, 2, etc.
Look at
http://www.maths.lancs.ac.uk/~gilbert/m243a/node6.html
 
Last edited by a moderator:
  • #10
Thankyou Bsmsmstmsphd but it's not the homework, I'm preparing for O'level matmatics exams so I need the method... thankyou
 
  • #11
Then, unfortunately for you, there is no single "method" for determining the general term of a sequence from some of its values. In fact, given any finite number of terms there exist an infinite number of different sequences taking on those values.

If "the simplest polynomial formula" is sufficient then Lagranges' formula or (equivalently) Newton's divided difference formula would work.
 

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