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

Triangular and Stellar Numbers

Extracts from this document...

Introduction

Math I.B Internal Assessment: SL Type 1

Stellar Numbers

Raj Devraj

Math I.B Internal Assessment: SL Type 1

Stellar Numbers

6/26/2011

St. Dominics International School

Raj Devraj


TRIANGULAR NUMBERS

TRIANGULAR NUMBERS WITH THREE MORE TERMS

image00.png

GENERAL STATEMENT: NTH TRIANGULAR NUMBERS IN TERMS OF N.

The differences between the sequences of terms:

image01.png

X

Y

Y= number of dots on triangle

X= number of dots on 1 side of triangle

0

0

1

1

2

3

3

5

According to Finite Differences if the 3rd difference of a pattern is 1, then the general term is a quadratic equation: image16.pngimage16.png
Thus to find the general term we must first find the value of ‘c’:
image12.png


In order to find the values of ‘a’ and ‘b’ we must solve a quadratic using simultaneous equations, thus:image18.pngimage17.png

Substitute the value of ‘a’ of one equation:image19.png

image20.png

...read more.

Middle

STELLAR NUMBERS

NUMBER OF DOTS TO S6 STAGE

S1

S2

S3

S4

S5

S6

1

13

37

73

121

181

Thus, using finite difference:

image02.png

The most obvious pattern is that the 1st row all numbers and odd and the second row all are even.

 Also all these numbers are some multiples of 12 + 1, for example: 12 also turns out to be the half of 6.

image03.png

6 STELLAR NUMBER AT STAGE S7

1 + 1 (12) + 2 (12) + 3 (12) + 4(12) + 5(12) + 6(12) = 253

GENERAL STATEMENT FOR 6 STELLAR NUMBER AT STAGE SN IN TERMS OF N

If you notice the multiples are triangular numbers thus the general statement will, in some way contain the general statement of the triangle numbers.  Because the final difference of the stellar numbers is 12 the multiple is Thus:

image04.png

4 STELLAR NUMBER

NUMBER OF DOTS AT 4 STELLAR

S1

S2

S3

S4

S5

S6

1

9

25

49

81

121

4 STELLAR NUMBER AT STAGE S7

1 + 1 (8) + 2 (8) + 3 (8) + 4(8) + 5(8) + 6(8)

...read more.

Conclusion

image14.png

The next difficulty faced is: if I multiply by the number of vertices the points on the closest will also be multiplied twice, and this could not be.

But then how do we add the inside star to the final result. In the example below the star on the inside has 12 dots (excluding the point in middle), and the outside star if you count the dots on the inside twice you will count a total of 12 dots, thus covering the dots of the inside star –

image15.png

                This is the same for all other stellar shapes.

...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. Extended Essay- Math

    �ٿ��(c)��'M�=�>�ne(c)n;�(��<�S��"R1`KE(c)��bz��k3/4�(c)�UNo�� ��)��i �Ҵh >(%��)�"�����1�0`"GN,2 � q1/4f�^01/2'%�\��ï¥<6�yRR"�Q-�mÛ¶s��]"v��[aÏ=�_�l(tm)<pb'Q�...+�1/4��nÛºu��/U'' G.1/4�}��-3/4J���KF '�|��"����� �"�������ܹsCCC�@-,��(��PFP58+��q"II�3�������&UΟR���j!��'ZZZ�v"|[b'Qjjj��D"'L&I(c)�"�c�֬Y��'P�3/4�PWT� "Q3/4��2Û~�E>��"����#��]w���""R�%VG8�'��"#�a%�L7��>���?�è£ï¿½R�'Ò+�3/4�� 6|�3�)��'�+��"S�Jb(tm)'*���o��s:���"��88"Ø´iÓ ï¿½""KB(c)�k uE��rb�QI,"(c)f��[o1/2�� -����"&S'k��0NI��- ��)00E�����...�-{�'G-�V�[�c b(tm)����3/4é¦ï¿½ï¿½ ,Z��"&P�[�>��8z��-��eIM>JJ�1/4Â-*�ZL����K1xIi�T*...#R��B�"yRR���0T3/4%P���)...��!�'%�"XXI$pJ��p� ��"()���J ��PR\"�oH��$@I1%V' \D�'�"|C$0%J�)��'H�""�� �) PRL�...�$@ �� ß LI�'bJ,�$��%�E8��H`J"Sba% ��E().�7$@S �� +I�."@Iq 3/4!�'%�"XXI$pJ��p� ��"�?k� �U�pIEND(r)B`�PK !��_"4V4Vword/media/image15.png�PNG IHDR"�K_p�RiCCPICC Profilex�YwTͲ�(tm)�,KXr�9�s"�a�K�Q��"� H ���*��""(�PD@�PD$1/2A?�{�y��ys����v���� �JTT�@xD\��(tm)!��" ?n�:� ����F��Y#��\����5&�#�0�72�@��������k��~Q1q

  2. Maths Internal Assessment -triangular and stellar numbers

    S1 S2 S3 S4 S5 S6 Layer 1 1 1 1 1 1 1 Layer 2 0 12 12 12 12 12 Layer 3 0 0 24 24 24 24 Layer 4 0 0 0 36 36 36 Layer 5 0 0 0 0 48 48 Layer 6 0 0

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

    Using this visual representation, we can model the logical pattern of the stellar shapes with three and four vertices: at each consecutive stage, the new outer simple star has two more dots per each vertices comparing to its predecessor. We can summarize this in the following table: n - Stage

  2. Stellar numbers

    12 12 2 13 24 12 3 37 36 4 73 The variables will be defined the same for tables to do with triangluar numbers: -n will be defined as the stage number of the stellar shape - as the nth stage of the stellar shape From the table it can be seen that there is no common difference ()

  1. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    This is also congruent with real life as there will be a natural point where the record will be almost impossible to beat; as there are natural forces such as gravity and friction that dictate the extent to which humans can jump; as jumping is contingent on one?s muscles take-off force; which also has a natural limit.

  2. Mathematic SL IA -Gold medal height (scored 16 out of 20)

    the solution is (remain to 6 significant figures); So, the cubic function model 2 is, y= Figure 9 The red points show original data, the blue curve is the cubic function curve. X Y cubic function model 2 Difference value (Y2-Y)

  1. Investigating Slopes Assessment

    For example, for case one I will use n=3. This means that my three other cases for case number 1 will be by substituting ?a? with 1,2 and 3. I will do then the same for n=4 and, if necessary, for n=5.

  2. Stellar numbers. This internal assessment has been written to embrace one of the ...

    is the total number of dots and the 1st and 2nd differences are self-explanatory.With a mere glance at the total number of dots, y, one might not notice the pattern immediately. The next aspect to be inspected is the 1st difference of y.

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