• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14

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

Extracts from this document...

Introduction

Stellar numbers Folio                                                                                                    Michael Wong

Introduction

In this folio task, we are going todetermine 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,

Side no.

Square no.

1

1

2

4

3

9

4

16

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.

...read more.

Middle

To see the graph, we can Select GRAPH again,

P (5)-STELLAR SHAPE

By using computer software “stellar”, we can plot the shapes below,

                            S5                                                         S6

Here are the first six stages of the P (5)-STELLAR SHAPE, and we can now create a table to demonstrate the number of dots in each of the stage,

n

1

2

3

4

5

6

Fn

1

11

31

61

101

151

One more layer will be adding on the privets stellar by increasing the general term number to 1. Thus, the different between F1 and Fn+1 will be the outer shell dots number (Fn). Every new lay been added on the stellar, its outer shell number will have 10 more dots.

Fn is the number of dots that the diagram contains.

When n=1, F1=1

n=2, F2=1+10×(2-1) =1+1×10

n=3, F3=1+10×(2-1)+10×(3-1)=1+3×10

n=4, F4=1+10×(2-1)+10×(3-1)+10*(4-1)=1+6×10

n=5, F5=1+10×(2-1)+10×(3-1)+10×(4-1)+10×(5-1)=1+10×10

n=6, F6=1+10×(2-1)+10×(3-1)+10×(4-1)+10×(5-1)+10×(6-1)=1+15×10 etc.

When n=n, Fn=1+X×10

X (the number that multiple to 10), has the following relationship with n when p=6

N

1

2

3

4

5

6

X

0

1

3

6

10

15

It also follows the general formula X =(n-12+n-1)/2

Try the pattern F7,

X=U7=7-12+7-12=21

When n = 7, F7 = 1+10×(2-1)+10×(3-1)+10×(4-1)+10×(5-1)+10×(6-1)+10×(7-1) = 1+21×10

          F7 = 210

Substitute X =(n-12+n-1)/2 into formula Fn=1+X×10,

Hence the general formula for Stellar number will be

Fn=1+10((n-1)2+n-1)2

=5n2-n+1

Now, test the formula by using n=4,
Fn=5n2-n+1

=542-4+1

=61

The F4 value is also same as the F4 value in Table on the page 7.

Again, check the answer by using the TI-84Puls calculator with the same method which is used for the triangular numbers.

...read more.

Conclusion

For example, p( 5 )-stellar shape as an example.

As the diagram show, although we had changed the density of the p (5)-stellar shape, they also contain 5 vertexes and actually contain the same number of dots.

The result is reliable, as the general statement is supported by the equations generated in the cases of a p (5), p (6), p (7) and p (8) -stellar shape. The numerical coefficients in each equation are equal to the corresponding stellar number.

However, all the general term number (n) must be greater than 1 (n≥1,n∈Z+). As a general term number, it can’t be 0, miner’s term and even decimal term.

Sn=pn2-pn+1

(p) Must be greater than 3 and applied to all p-stellar shapes given that pϵ R.


P.T.O. For Appendix.

The methods of use TI-84 Plus calculator to find out the relationship between general term number n and the triangular number (number of dots) in a general statement.

  1. Press STAT then ENTER and choose “1.Edit”, (P.T.O to see) From the calculator screen,
  2. Type in General term number (n)into “L1” and Triangular number (Un) into “L2” in the order as the diagram shown (P.T.O to see) below. And then press2ND and MODE to choose “QUIT”.
  1. Press STAT then to “CALC”, and chose “5. QuadReg”. Finally, press “ENTER” to get the result in diagram in the right hand side.
  • This method can also use on finding the general formula of the special number other geometric shapes Sn, Kn, Fn or Mn for L2 instead of Un.
...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. Maths Internal Assessment -triangular and stellar numbers

    The chosen value for p is 7. This means that we are exploring the 7-stellar number therefore, the star has 7 points. Question 6.1: Find the number of dots (i.e. the stellar numbers) in each stage up to S6. Organize the data so that you can recognize and describe any patterns.

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

    Question 3 Consider stellar (star) shapes with p verticies, leading to p-stellar numbers. The first four representations for a star with six verticies are shown in the four stages S1-S4 below. The 6-stellar number at each stage is the total number of dots in the diagram.

  1. In this task, we are going to show how any two vectors are at ...

    Another graph can be drawn below. Figure 5 Also, the slope of these two point is same as the slope of the equation . (Refer to figure 1) Again, by using the diagram Figure 6 We can find out that the formula t ? 0 is also fix on this two point and from the vector equation:.

  1. Stellar Numbers. In this task geometric shapes which lead to special numbers ...

    Consider stellar (star) shapes with p vertices, leading to p-stellar numbers. The first four representations for a star with six vertices are shown in the four stages S1-S4. The 6-stellar number at each stage is the total number of dots in the diagram. 3. Find the number of dots (i.e.

  2. Comparing the surface area of different shapes with the same volume

    The use of this algebraic expressions to find h in terms of r for the cylinder and H in terms of a for the prism. A) 1000cm3= ?r2 x h h= 1000cm3/ ?r2 Surface Area of Cylinder with imputed h Abpcc = ?r2 + 2?rh Abpcc = ?r2 + 2?r (1000cm3/ ?r2)

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

    Using the expression one can continue the triangular number sequence indefinitely. Task 2 Find a general statement that represents the nth triangular number in terms of n. In the previous task, we defined the nth triangular number in terms of the previous triangular number In this task we can derive

  2. Stellar numbers and triangular numbers. Find an expression for the 6-stellar number at ...

    If we now turn our attention to stellar (star) shapes, these have p vertices which then lead onto "p-stellar numbers". If you look at the first four examples below, we can see stars with six vertices shown in stages s1 - s4.

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