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Stellar numbers. The aim of the current investigation is to consider different geometric shapes, which lead to specific numbers, to formulate the universal formulas

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Introduction

Miras International School Almaty 2012

INVESTIGATION

Subject: Mathematics SL

Topic: Stellar Numbers

Name: Elena Arkhangelskaya

Grade: 11 IB

Date: 26th of March

Mathematics SL Investigation

Stellar Numbers

A number sequence is a list of numbers where there is a pattern.[1] We study variations of different sequences and series, such as: geometric, arithmetic and others. We learn different formulas to find the unknown values of the term number, the sum of the terms. And the only reason we do that is to make our live easier and convert huge sequences of numbers into short and exact formulas. In the current investigation I’m going to analyze different kinds of the sequences, which include not only numbers but geometric shapes and stellar numbers as well.

Aim

The aim of the current investigation is to consider different geometric shapes, which lead to specific numbers, to formulate the universal formulas to every specific group of geometric shapes and to test the validity of the gotten general statement.

Procedure

...read more.

Middle

n = 5

u₅ = 15

u₅ = u₄ + n = 15

n = 6

u₆ = 21

u₆ = u₅ + n = 21

n = 7

u₇ = 28

u₇ = u₆ + n = 28

n = 8

u₈ = 36

u₈ = u₇ + n = 36

The general statement that represents the nth triangular number in terms of n is un  = un-1   + n

  1. The second task is to consider stellar (star) shapes with p vertices, leading to p-stellar numbers. Stages S₁-S₄ represent the first four stages for the stars with six vertices.

image03.pngimage04.pngimage13.pngimage08.png

image00.png

            S₁                            S₂                                      S₃                                                      S₄        

  • To find the number of dots in each stage up to S₆.

Term Number

Number of Dots

n = 1

S₁ = 1

n = 2

S₂ = 2(n-1) 6 + S₁ = 12

n = 3

S₃ = 2(n-1) 6 + S₂ = 72

n = 4

S₄ = 2(n-1) 6 + S₃ = 120

n = 5

S₅ = 2(n-1) 6 + S₄ = 180

n = 6

S₆ = 2(n-1) 6 + S₅ = 252

  • To find an expression for the 6-stellar number at stage S₇.

S₇ = 2(7-1) 6 + S₆

  • To find a general statement for the 6-stellar number at stage S  in terms of n

S  = 2(n-1) 6 + S

  • To repeat the steps above for other values of p (p=5; p=7)
...read more.

Conclusion

. 2 is multiplied by the

(n-1), which means that the number of dots on the each ray is subtracted by 1, because the internal vertices of the star are divided into two rays as well, so that the number of dots is not surplus. 2(n-1) is multiplied by the number of vertices (p) and the previous stage of the sequence (S      ) is added.

Conclusion

The aim of the investigation was to consider different geometric shapes, which lead to specific numbers, to formulate the general statements to every specific group of geometric shapes and to test the validity of the gotten formula. By considering different types of geometric shapes such as triangular numbers, 6-vertices stellar shapes, 5-vertices stellar shapes and 7-stellar shapes, we proved that the use of arithmetic and geometric sequences is not limited by only working with numbers, different geometric forms can be measures as well. The general statement S  = 2(n-1) p + S     was produced, and also tested on validity and limitations.

Bibliography

  • Mathematics SL electronic book
  • Stellar Numbers Task Sheet. For final assessment in 2011 and 2012


[1] Math SL HaH Text

...read more.

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