Stellar Numbers. In this folio task, we are going to determine difference geometric shapes, which lead to special numbers.

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Stellar numbers Folio                                                                                                    Michael Wong


In this folio task, we are going to determine difference geometric shapes, which lead to special numbers. The task will be achieved by using the theories of sequences and series. Due to the difference pattern of stellar numbers diagrams, the first thing to do is drawing diagrams and hence to work out the change of the dots and explain the pattern for each of them. After that, we are able to find out the general term of them and test them by substituting a number. Also, TI-84 Plus calculator will help to prove the statement is correct or not.  The task will be similarly achieved by the example below,

By seeing the pattern, from 0, there is no square without any dots. When the general number changed to 1, there is 1 dot.

In the following, the general term number changed to 2, there are 4 dots and hence to create a square.

Furthermore, when the general number changed to 3, there are 9 dots which will create a bigger size of square.

Hence, we are able to create a table to demonstrating the change rule of the square stellar diagram below,

Assume, Un = Sq. no. (Square number)

n = side no. (gerneral term number)

Therefore, the general statement of it is Un = n2,         for      ( n: n ϵ Z+) (n≥1)

To test this general statement, by subsidising 2 into n, so U2=42, hence we can prove that when square number is 2, it has 4 dots.  Finally, check it with the TI-84 calculator.

After that, we could consider the special numbers for the geometric shapes in the task.

Results/ Analysis

Now, a diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each of them are (1, 3, 6, 10 and 15) and those numbers can be known as the square numbers.

By seeing the diagram, we can know that one more row of dots will be added by the general term number increase 1, the new row will have 1 more dots then the pervious row. Hence, the total row number and dots in the new row is same as the number of the general term.

If we now are going to find out three more terms of the triangular numbers, triangular numbers are known as "the number of dots in an equilateral triangle uniformly filled with dots", and it can be shown below by drawing from the computer.

From the diagram, the three diagrams have black dots and white. The black dots are the original dots starts from the square number Sn=15. And the white dots are dots had added by increasing the general term number. Thus, we are now able to create a table of demonstrating the side numbers and Square numbers as the following.

Consider about the triangular number, we can work out,

When n=1, U1=1=1×1

n=2, U2=3 =6÷2= (2×2+2)÷2

n=3, U3=6=12÷2= (3×3+3)÷2

n=4, U4=10=20÷2= (4×4+4)÷2

n=5, U5= 15=30÷2 = (5×5+5)÷2

Therefore, Un=(n2+n)÷2

For testing the formal, we could substitute a random number into it,

Assume n=7.

Un= (7×7+7)/2=28

It is same as the result referring to the diagram.

Now, we could also check the answer again with the TI-84Puls calculator. (Methods will show Appendix 1 in the end of the portfolio.)

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The results demonstrated from the calculator is y=0.5x2+0.5x+0

So,                  y= 12n2+ 12n=12n2+n

y= n2+n2                      ,    y=Un,       x=n


As the general term number can’t be zero or miners, the final general statement will be:

Un=(n2+n)÷2           for             n:n∈Z+   (n≥1)

Also, we can see a graph of the numbers by Select GRAPH,

Now we are able to see the graph which is plotted by the side no. and square no.


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