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Maths SL, Type 1 Portfolio - triangular numbers

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Introduction

Maths Practice Portfolio

Maths Portfolio Type I

Special numbers go back in history and there is a great relation between the theorists and the maths they discovered. They are numbers with unique properties, making them different to other ordinary numbers. ‘The origins of the concept of the shape of number’ is a topic which can be directly related to this fact. The idea behind this is that there are many origins of the concept of the shape of number. In the following task we were investigating patterns in geometric shapes that will lead to the formation of special numbers. More specifically, we will look at triangular patterns that will enable us to discover a pattern of special numbers. The first part of the investigation we looked at a triangular pattern formed with dots in the shape of triangles to and calculates the nth term for this pattern. Original Sequence

Counting the number of dots in each of the triangles, we can see that there is a pattern. The numbers of dots increase by (n+1) adding 2, 3, 4 and 5. Therefore, this hints that the next three terms will be as we will be adding 6, 7, 8.

Middle

From the above table, we can see that our variables are n and Tn. When we try to classify this pattern, we can see that it is not arithmetic. Arithmetic sequences require a common difference (d). Meaning that when we subtract from we should get the same value each time      This continues so that the value in between each increases by 1. The common difference is not equal we do not have an arithmetic sequence. The sequence is also not geometric as it would have a common ratio (r). If we were to have a geometric series when we divide the value of r will equal.    So, in our case, this particular triangular pattern sequence is not considered in the above two categories. This sequence can be categorized in the ‘special sequences’ which consists of different series such as triangular numbers, square numbers, cube numbers, Fibonacci numbers and more. One of the reasons why this sequence is called triangular numbers is because if the sequence is physically drawn, the shape of the diagram would be a triangle with dots equally spaced out, hence the name ‘triangular numbers’.

Conclusion      Substituting 7 into the formula to replace 6:

6n(n-1) +1

7n(n-1) + 1

When n = 1

7(0) = 1 = 1

n = 2

14(1) + 1 = 15

This was tested for all the other 5 terms and was found to be correct.

Next, we tried a 5 stellar shape.      81

Substituting 5 into the formula to replace 6:

6n(n-1) +1

5n(n-1) + 1

When n = 1

3(0) +1 = 1

n = 2

5(2)(2-1) + 1 = 11

This was tested for all the other 5 terms and was found to be correct.

Therefore the general statement is pn(n-1) + 1

To arrive at this formula, we noticed that p was the stellar star shape and when n was put into the formula, we would be able to achieve the values required. However as this is a stellar star shape, we cannot calculate the series for a stellar star shape that is less than three. Therefore p>=3 in order for the formula to work.

In conclusion, we can say that geometric shapes lead to special numbers.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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