Polygonal Numbers Essay

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Polygonal Numbers Table of Contents Introduction Pg.3-4 Formulas for finding the nth polygonal number 5-25 Formulas for finding the nth sum of polygonal numbers27-30 Formulas for finding the relationship between polygonal numbers 31-36 Appendix Pg.37-38 Conclusion Pg.39-40 Introduction I was in math class when my brilliant math teacher had out a worksheet on triangular numbers. I was presented with the diagram on triangular numbers. N represented the number of dots on one side and Tn represented the total number of dots in a triangular array. Below is a chart that represents the question. My goal was to figure out the 5th, 6th, 7th number and so on and to come up with a formula in finding the nth triangular number. Of course, I couldn’t keep drawing diagrams of triangular dot arrays forever and count the total dots even when the n was a very large number, say 100. It would just be tedious and very time consuming so I decided to come up with a logical and correct way to approach such a problem. One of my first observations was that each nth triangular number was Tn plus N. This observation would most likely was going to be helpful in finding the answer. N | Tn | 1 | 1 | 2 | 3 | 3 | 6 | 4 | 10 | … | ? | I found this question to be really intriguing and I was determined to find an answer. I realized that this was a sequence-related type of question and in many previous instances; sequence-related questions often appeared in which case I never knew how to solve them. I decided that if I further investigated this question and related questions, I would be able to for once and all understand, interpret and answer such questions. I realized that not only could triangles be investigated but also other polygons. Thus, I determined that I was going to be investigating polygonal numbers. On the way, I hope to come up with
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