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I am interested in learning more about patterns.

My objective is to learn about how to analyse a set of numbers (not infinite but a large set of numbers in sequence) and see the patterns and then write them down and use that pattern for something else in the future.

What should I need to do? Where do I start etc...

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Do you know the game 'GO'? –  Berci Jan 31 '13 at 15:04
    
Ok; just begin: 1,2,3,4,5,6, find next term. –  user55514 Jan 31 '13 at 15:16
    
This question may not be specific enough. What kind of numbers would we be looking at, and what kind of math do you want to apply these to? If you really want a place to start, you could try some IQ tests that have numbers. –  TakeS Jan 31 '13 at 15:45
    
Thank you all you guys, I realised it's a vague question, I was hoping where would I go about to research myself, or find materials to learn about patterns in number sequence. I tagged soft-question here which does suggest the question may not require a definitive answer. thanks @amzoti your answer is very useful, if I could be a little bit pushy, do you know where I can find any resources to learn about "patterns in a sequence of numbers" –  Val Feb 5 '13 at 9:23
    
@Val Your question is probably appropriate for the nearly-in-beta-SE area51.stackexchange.com/proposals/64216/…. Check out the proposal and commit to it if you're interested. Then we can get it off the ground and get the site in beta! –  Xoque55 Mar 3 at 4:52
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2 Answers

Your questions is rather broad, as we can have all types of patterns that can include logic patterns, number patterns, and even word patterns.

For number patterns, there are all sorts of things (list not exhaustive) to learn, investigate and explore, such as:

$\bullet$ Arithmetic Sequences

$\bullet$ Geometric Sequences

See for example, the Number Sequence Calculator

$\bullet$ Special Sequences

$\bullet$ Square Numbers, Cube Numbers, Fibonacci Numbers, Pascal's Triangle, $\cdots$

$\bullet$ General Number Patterns

$\bullet$ Repeating Patterns

$\bullet$ Recursions

$\bullet$ Method of Common Differences

$\bullet$ Non-Math Sequences

$\bullet$ Mathematical Series

$\bullet$ Discrete Mathematics

Images

There are also very interesting patterns and here are some images of those.

There are lots of general techniques like taking the first difference of members (works for quadratic equations, linear recurrence relations, etc.) or looking at subsequences (say, every other member).

You can browse through the OEIS and see the many types of relationships you can have (from the very simple to the very complex).

This site has over 400 examples of Number Patterns and look at the magic anything graphics and patterns.

There are entire books written for all kinds of patterns and symmetry within mathematics.

Regards

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(+1): Your answer is very well researched. –  Parth Kohli Jan 31 '13 at 16:19
    
@Novice: Thank you! Regards –  Amzoti Jan 31 '13 at 16:21
    
I agree with Parth!! $\;\;\Large \ddot\smile\;\;$ –  amWhy May 5 '13 at 0:20
    
@amWhy: This is one the very beautiful areas of mathematics, the search for pattern and then mathematical methods to generalize, study and catalog it. Thanks. –  Amzoti May 5 '13 at 0:51
    
I agree, entirely...that's a large reason I love math...and learning in general...how are things related? connected? interrelationships? recurring patterns?...etc. –  amWhy May 5 '13 at 0:54
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There are patterns and they can be found. But you might also be able to choose the number you want. For instance; next term after 1,2,3,4,5,6, is obviously 1000, because the general term of the sequence might be :

$a(n)= n + ((n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(a-7))/6! $

and then you have just to choose the value of a you wish for it to be next term.

Examples of apparent patterns that eventually fail

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