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Triangular and Stellar Numbers

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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.

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