The table below organizes the data so that we can recognise and describe the patterns.

Excluding the first layer (layer 1), as the stage number (Sn) and layer increases by 1, the number of dots correlatively increase by 12.

Question 4:

Find an expression for the 6-stellar number at S7.

As we discovered the connection between the number of layers and the number of dots, an expression for S7 can be calculated.

Therefore, the expression for the 6-stellar number at stage S7 is (12 x 21) +1.

Question 5:

Find a general statement for the 6-stellar number at stage Sn in terms of n.

Similar to using square numbers to find out triangular numbers, finding the general statement for the 6-stellar number will require the manipulation of triangular numbers.

Firstly we shall find more expressions for the 6-stellar numbers:

When the expressions are written this way, we can relate the 6-stellar numbers with triangular numbers. The focus is on the circled number in the expression as they correspond to the n-1th of the triangular term. In other words the general statement for the 6-stellar numbers is (12 x (n - 1)) + 1.

However, this expression is not complete as it does not apply to the 6-stellar numbers. Therefore, as previously stated, for the triangular numbers, n-1 = n(n-1)/2. So if this is substituted into the equation, the general statement should be discovered.

Therefore, the general statement for the 6-stellar numbers is 12 x ((n(n-1)/2)+1

Question 6:

Now repeat the steps above for other values of p.

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.

Excluding the first layer (layer 1), as the stage number (Sn) and layer increases by 1, the number of dots correlatively increase by 14.

Question 6.2:

Find an expression for the 6-stellar number at S7.

Therefore, the expression for the 7-stellar number at stage S7 is (14 x 21)+1.

Question 6.3:

Find a general statement for the 7-stellar number at stage Sn in terms of n.

We must verify if the same theory, behind the general statement for the 6-stellar number, can be used to determine the general statement for

the 7-stellar number. This means that the manipulation of triangular numbers is essential.

Therefore, from the examples above, we can see that the same logic from the previous statement (6-stellar number) can be used. For example, the expression for the 7-stellar number at S4 is S4 = (14 x 6) + 1

It should be recognised that the '6' (circled in the calculations above) is the 3rd triangular number. Thus, there is the same logic behind this general statement as the previous general statement (6-stellar number). The general statement of the 6 - stellar number was 12 x ((n(n-1)/2)+1. There was 12 more dots added onto each sequential layer and the equation commenced with 12 x… However for the 7 - stellar number, 14 more dots are added onto each sequential layer.

By taking this into account, the general statement for the 7 - stellar number is 14 x ((n(n-1)/2) + 1.

Therefore, the general statement for the 7 - stellar number is 14 x ((n(n-1)/2 )+ 1.

Question 7:

Hence, produce the general statement in terms of p and n, that generates the sequence of p-stellar numbers for any value of p at stage Sn

Sn for the 6-stellar numbers is 12 x ((n(n-1)/2 )+ 1.

Sn for the 7-stellar numbers is 14 x ((n(n-1)/2 )+ 1.

However, the question is asking for a general statement that generates the sequence of p-stellar numbers for any value of p. Thus, when we compare these two statements; we can see that the only integer which changes is the first, and hence, finding its mathematical relation between the first integer and its corresponding 'p' value is necessary to form this general statement.

The 'p' value is the number of points on a star or the p-stellar number.

Therefore, in this case, the 'p' values are 6 and 7.

The first integer for the 6 - stellar number (p) is 12.

The first integer for the 7 - stellar number (p) is 14.

We can see that the first integer is double its 'p' value.

6 x 2 = 12

7 x 2 = 14

Therefore, the general statement in terms of p and n is:

Sn = pn (n-1) + 1

However, when the relationship between the total number of dots and the layers are plotted onto a graph, a new formula is formed.

A polynomial trend-line is used to determine the link between each layer to determine the formula for the 6-stellar number. This formula is y=6x2 – 6x + 1. The 6 in ‘6x2’ and ‘6x’ from the n-stellar number. Therefore, if it was a 7-stellar number, the equation should be y = 7x2 – 7x +1; which it is shown in the graph below. If this equation is expressed in terms of ‘p’ and ‘n’, it will be pn2 – pn + 1. This simply justifies the general statement previously found which was: Sn = pn (n-1) + 1 as the statement from this graph is just the expanded version of the found general statement. The ‘R’ value is the accuracy of the plotted data when compared to the polynomial trend-line. This value state 1 which means that it is 100% accuracy which supports the equation by stating that it is 100% accurate.

The graphs below display the other values of ‘p’ to test the validity of this equation.

Question 8:

Test the validity of the general statement

Question 9

Discuss the scope and limitations of this general statement

A few limitations come with the use of this general statement.

There are three positive numbers that cannot be used in this general formula for the values of 'p'. They are: 0, 1 and 2. The simple reason is because stellar-numbers are presented in the shape of a star. Therefore as the 'p' value represents the number of points on a star, - e.g. for a '2-stellar', this would take the form of a straight line which does not fulfil the definition of 'stellar number'.

However, as stellar numbers must take the form of a star; p = (infinity) cannot be applied to the general statement as the shape would be a circle.

As stated in the previous limitation, that p= (infinity) will be a circle; this indicates that the higher the 'p' value is; the harder it will be to create that 'n- stellar number' and thus harder to prove.

With logical reasoning, it is obvious that no decimal numbers can fill the 'p' value when creating the diagram for it. There is no such thing as 'half-a-dot' (p+0.5) nor 'a-quarter' (p+0.25). Although these numbers can be substituted into 'p' when working out how many dots are in that stellar-number, there is no way to calculate if the number of dots is correct. Therefore, a 'decimal numbered stellar' cannot be created.

Similar to the reason behind why decimal numbers cannot be a ‘p’ value of the stellar number, negative numbers cannot take the value of ‘p’ because there is no such thing as a negative dot. It would simply mean that there is no dot.

E.g. -3 Stellar, layer 3

Sn = pn (n-1) + 1

Sn = ((-3)x(3))(3-1) + 1

Sn = (-9 ) x (2) + 1

Sn = -18 + 1

Sn = -17

= drawing -17 dots is impossible.

Question 10:

Explain how you arrived at the general statement.

The results from a few different techniques were required to arrive and form the general statement. Firstly, it started with the triangular number formula which was supported by the geometric reasoning behind the formation of square numbers. It was essential to understand and explore these triangular numbers as without this, there would be the need to write out the sequential sum for the further layers.

Secondly, the knowledge of the triangular shapes was also applied to the creation of the 6-stellar numbers to discover the connection between the number of dots and the layers. By finding this link, it was possible to create an expression for the 6-stellar number which linked with the triangular numbers. All these previous steps led to the formation of the general formula which was 12 x ((n(n-1)/2)+1.

Then, by drawing the diagram of the 6-stellar number a particular pattern emerged. It was about the consistent addition in the number of dots as the layers increased. In other words, the first 'star' consisted of one dot, the second layer (star) contained 12 dots, the third had 24 dots, the fourth layer had 36 dots. As one can see, the number of dots is increasing by 12 excluding the first -> second layer. Without exploring the triangular numbers, it would have been difficult to express this abnormality.

Fourthly, the relation between the 'p' value and the first integer of the general formula must be determined. This was found by comparing the 6-stellar and 7-stellar numbers, and was found that the first integer was always equal to 2 x p.

As all of the steps above was completed, what one needed to do now was just combine the results from the steps above to determine the general statement. Therefore, by doing so the general statement was found to be: Sn = pn (n-1) + 1.

The final step was to test if the general statement could be applied to the stellar numbers in terms of 'p' and 'n'; which was successful; however with some few limitations such as the 0, 1, 2 and decimal numbers substituting the 'p' value.