Mathematics of Patterns

Patterns are consistent and recurring sequences and can be found in sets of numbers, events, shapes, nature and almost everyplace you care to look. Examples of patterns include seeds in a sunflower, geometric designs on quilts, and the number sequence 0;3;6;9;12;….
In a number pattern, the following notation should be used:
The 1st term in a sequence is T1
The 5th term in a sequence is T5.
The 9th term in a sequence is T9.
The general term, nth, is written as Tn. If a sequence follows a pattern, you can calculate any term by using the general formulae. Therefore, if you can find the relationship between the term’s position and its value, you will be able to describe the pattern and discover any term in the sequence.
Some sequences have a constant difference between two successive terms. This phenomenon is referred to as a common difference. The common difference is often denoted by d. For example, in sequence 10; 7; 4; 1;…, the common difference is 3. To find the common difference, the difference between two successive two must be known (d=T2-T1)
The difference between Tn and n should be noted. N is like place holder that indicates the position of the term in a given sequence. On the other hand, Tn is the value of the place that is held by n.
References
https://www.siyavula.com/maths/grade-10/04-number-patterns/04-number-patterns-01.cnxmlplus
https://www.learner.org/teacherslab/math/patterns/number.html

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