• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Maths SL, Type 1 Portfolio - triangular numbers

Extracts from this document...

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.

image31.png

                                        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.

...read more.

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 image00.png

 from image01.png

 we should get the same value each time

image02.png

image03.png

image04.png

image05.png

image06.png

image07.png

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 image08.png

 the value of r will equal.

image09.png

image10.png

image11.png

image12.png

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’.

...read more.

Conclusion

image20.png

image21.png

image22.png

image23.png

image24.png

image25.png

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.

image20.png

image26.png

image27.png

image28.png

image29.png

image30.png

 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.

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Math IB HL math portfolio type II. Deduce the formula Sn = ...

    So: a5 + a36 = 8 + 32 = a12 + a29 = Because: 5 + 36 = 40 = 12 + 29 So: a12 + a29 = 40 So: a39 + a2 = a30 + a11 = a12 + a29 = 40 Because: 39 + 2 = 30 +

  2. The Fibonacci numbers and the golden ratio

    1,618056 13 233 1,618026 14 377 1,618037 15 610 1,618033 16 987 1,618034 17 1597 1,618034 18 2584 1,618034 19 4181 1,618034 20 6765 0 By investigating the Fibonacci numbers, I can make a conjecture, that the ratio of two consecutive terms gets closer, as they increase, to the Golden Ratio.

  1. Math IA type 2. In this task I will be investigating Probabilities and investigating ...

    therefore we can conclude that Adam normally wins the practice games as he almost certainty gets more than half the points, which is more than 5 points. Part 2 Now I will look at Non extended play games where to win a game, the player must win with at least

  2. Maths Internal Assessment -triangular and stellar numbers

    This formula is y=6x2 - 6x + 1. The 6 in '6x2' and '6x' from the n-stellar number. Therefore, if it was a 7-stellar number, the equation should be y = 7x2 - 7x +1; which it is shown in the graph below.

  1. Stellar Numbers Portfolio. The simplest example of these is square numbers, but over the ...

    171 243 Question 4 Find an expression for the 6-stellar number at stage S7 The expression which I used for the 6-stellar number at stage S7 is carried down from the . previous information determined that each term is 12n more than the previous term, in which n is equal to the term number of the previous term.

  2. Math Portfolio: trigonometry investigation (circle trig)

    When the value of y is divided by the value of x, a negative number is divided by a negative number resulting to a positive number. Quadrant IV The value of y equals a negative number in the quadrant 4 and the value of r equals a positive number as mentioned beforehand.

  1. Stellar Numbers. In this study, we analyze geometrical shapes, which lead to special numbers. ...

    Judging from the table 3.1, we have the following result of the calculations: n - stage number - Stellar number (number of dots)

  2. Math Portfolio Type 2 - PATTERNS FROM COMPLEX NUMBERS

    Factorize for n = 3, 4 and 5. -1= (z-1)( -1= (z-1)( -1= (z-1)( 1. Use graphing software to test your conjecture for some more values of n ? Z+ and make modifications to your conjecture if necessary. Lets try it for n=6 Testing for Aw distance ; a=cis0 and w is cis120 =1.732050? ?17.3 (because of accuracy of software)

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work