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Stellar Numbers

Extracts from this document...

Introduction

IB DP Student

IB Standard Level Mathematics

Internal Assessment Type I

Stellar Numbers


Content:

Part I: Introduction                                                                 3

Part II: Triangular Numbers                                                         3

Part III: General Statement                                                        7

Part IV: 6-Stellar Numbers                                                         10

IV.a investigation                                                        10

VI.b Testing validity                                                        11

Part V: 5-Stellar Numbers

V.a Investigation                                                        12

        V.b testing  validity                                                        13

Part VI: General Statement for Stellar Numbers 12

IV.a Formation 16

IV.b Testing 17

Part VII: Conclusion 19


Part I:                Introduction

I was given the sequence of triangular numbers, the first five terms are shown to use. I was asked to list out the next three terms. First, I needed to determine the pattern. The purpose of this to let me be familiar with geometric shapes, so that I can understand them better, making it easier for me to accomplish my ultimate goal, finding a universal general statement for stellar numbers.

Part II:                Exploration of problem

        In this section I will explore and investigate the geometric shape given to me, the triangular numbers. Triangular numbers are numbers that can fill up an equilateral triangle. The dots in the triangle have to be equally spaced. My work in this section can help me understand and get familiar with not only the ways that triangular numbers work but also geometric shape and their ways of working, this will help with my investigation.

The given triangular number terms are:

image00.png

image00.pngimage00.pngimage00.pngimage00.png

...read more.

Middle

8

36

1+2+3+4+5+6+7+8

8

35

Part III:        The general statement for triangular numbers

III.a        Finding the general statement

        Next, I need to generate a gerenal statement, the general statement that represents the nth triangular number in terms of n.

        Since I am finding the nth term, then the equation would be the following shown below:

T(n) = 1 + 2 + 3 + 4 +...+ n                                (1)

        Then, I wrote a conversed equation of (1), shown below:

                                T(n) = n + n-1 + n-2 + n-3 +…+1                                (2)

        The reason for creating a conversed equation, is because after adding both equations together, a common difference or ratio, namely a relationship between the terms. After combining (1) and (2) together:

T(n) = [1+n] + [2+(n-1)] + [3+(n-2)] + ... + [(n-1)+2] + [n+1]                 (3)

         I get (3), and after simplifying (3), I got this:

T(n) + T(n) = (n+1) + (n+1) + (n+1) + … + (n+1) + (n+1)                

T(n) + T(n) = n(n+1)                                

T(n)  =                image04.png                                 (4)

         Thus, I have successfully generated a general statement in the form of n.

III.b        Testing the general statement

        In this section, I have to test my general statement to insure that the statement fits the sequence just because of coincidence. I’m going to use my general statement to find the 9th and 10th triangular number.

        Now I am going to find the 9th triangular number:image04.png

                                        T(n) = image01.png

T(9) =image02.png

 T(9) = image03.png

T(9) =

 T(9) = 45

        Now I am going to find the 10th triangular number:

image04.png

T(n) =image05.png

T(10) =

T(10) =

T(10) =

T(10) = 55

...read more.

Conclusion

I think that my general statement is the best way to calculate the terms, since it is easy to use and it is a lot quicker. Counting manually will be very time-consuming and a lot if the vectors is very large, and the term is very big there is a chance that I will count it wrong, since it is very easy to overlook some of the dots, and if you think that you missed something, then you have to start all over again. So it is better to use my general statement.

During the investigation, I used different kinds of technology to help me, including paint for the diagrams, math type for the equations. It is important to use different software to help because it can help show how technology can be part of real life; these programs can help solve mathematical problems. These program help speed up my investigation, and I will definitely use them again.

...read more.

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