# Stellar Numbers Portfolio

MATHEMATICS SL
INTERNAL ASSESSMENT TYPE 1
STELLAR NUMBERS

In this Internal Assessment, triangular and stellar number patterns were treated with a thorough investigation.

Triangular Numbers

To begin with, like square number patterns (such as 1, 4, 9, 16) triangular number patterns work in a similar manner. The first five triangular numbers are:

1                   3                   6                            10                        15

Following the same arrangement pattern of the dots within the triangular pattern, the 6th, 7th, and 8th triangular numbers are as follows:

21                                    28                                   36

An instantly noticeable aspect of the pattern is that from the triangular numbers 1 to 3, 2 is added; from 3 to 6, 3 is added; 6 to 10, 4 is added; from 10 to 15, 5 is added; from 15 to 21, 6 is added; from 21 to 28, 7 is added; and from 28 to 36, 8 is added.

From the above triangular number pattern, it is possible to deduce a general statement that represents the nth triangular number in terms on n.

Sn=1+2+3+…+(n-2)+(n-1)+n
+

Sn=n+(n-1)+(n-2)+…+3+2+1______________________

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

2Sn=(n+1)+(n+1)+(n+1)+…+(n+1)+(n+1)+(n+1)

2Sn=n(n+1)

Sn=

So, the general statement that represents the nth triangular number in terms of n is Sn=. In order to test the validity of this statement, let us insert the number 7 for n in an effort to find the 7th triangular number, which according to our previous data, is 28.

Sn=

S7=
S
7=

S7=

S7=28

Thus far, the general statement for the nth triangular number holds true. In order to ensure the general statement is completely correct, let us substitute the number 3 for n. The answer, according to the pervious data should be 6.

Sn=

S3=
S
3=

S3=

S3=6

Once again, the general statement Sn= stands its ground.

Stellar Numbers

Furthermore, stellar, or star, numbers also have a similarly executed pattern. For further purposes, p will represent the number of vertices a star, so it will have p-stellar numbers.

In the following representation of a star with 6 vertices, the number of dots represents the 6-stellar number. Therefore, the first four 6-stellar numbers are as follows:

S1                S2                            S3                                                S4

1                13                            37                                                   73

Following the same arrangement pattern of the stars and dots within the stellar pattern, the 5thth, and 6th 6-stellar numbers are shown below, labeled S5 and S6, respectively:

S5                                                               S6

121                                                             181

In the 6-stellar number pattern, it is observable that from 1 to 13, 12 is added; 13 to 37, 24 is added; 37 to 73, 36 is added; 73 to 121, 48 is added; 121 to 181, 60 is added.

From this information, it is possible to formulate an expression for the 6-stellar number at stage S7.

S7=1+12+2(12)+3(12)+4(12)+5(12)+6(12)

S7=1+12+24+36+48+60+72

S7=253

Moreover, using the same steps utilized for finding the general statement of the triangular pattern, a general statement for the 6-stellar number at stage Sn in terms of n can be fabricated.

Sn=1+12+2(12)+…+(n-2)12+(n-1)12+12n
+

Sn=12n+(n-1)12+(n-2)12+…+2(12)+12+1_______________________________________

2Sn=(1+12n)+(12+(n-1)12)+(2(12)+(n-2)12)+…+((n-2)12+2(12))+((n-1)12+12)+(12n+1)

2Sn=(1+12n)+(12+12n-12)+(24+12n-24)+…+(12n-24+24)+(12n-12+12)+(12n+1)

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