via Jain 108 Academy ------- POLYGON SEQUENCES:

Shapes That Generate Progressively Increasing Number Codes

A polygon is a shape with many sides, it could 3 or 4 or 5 or 6 or more sides. In the same fashion that we can build a sequence of Squared Numbers, like 1-4-9-16-25 etc we can build pentagonal, hexagonal, heptagonal sequences that carry on forever… Sometimes we need to know a certain number in a certain sequence at a certain position. eg: If I want to know what is the 4th Pentagonal Number, (the Sequence is 1-5-12-22-35-etc) there exists a special formula that can tell us this. It “t” is the number of sides in a polygon, (in this example t=5), the formula for the nth t-gonal number Poly(t,n) is [n^2(t-2)-n(t-4)]/2. Now substitute the values of t=5 and n=4 into this formula, giving: [4^2(5-2)-4(5-4)]/2 = [16x3-4x1]/2 = [48-4]/2 = 44/2 = 22 We just established that the 4th Pentagonal Number is 22 without knowing the full sequence of 1-5-12-22-35… This type of information for Pattern Hunters who need to know such data for their research. Jain 108 Image: (thanks to Tony Foster, who runs a Facebook Group called Pascals Triangle, that has over 6,000 members, for submitting this artwork on his site)

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