y= 12x2+ 12x
The general statement has to be in terms of n, not x¸ but beforehand, this equation can be further simplified. The common factor of 12 can be factored out:
y= 12(x2+ x)
whichcan be rewritten as division rather than multiplication:
y=(x2+x2)
Since x is equivalent to nas they represent the same variable, therefore x=n,n is substituted in the equation, thus the general statement regarding triangular numbers is formed:
wheren∈N.y represents the total number of dots and n represents the term of the triangular number.
Table 2
However, this is not the only way such a statement can be generated. The pattern of the sum of y shown in Table 2 can be represented by the following summation formula:
y=i=1ni=1+2+3+…+n-1+n
or
y=i=1ni=n+n-1+n-2+…+2+1
Chart 1
The two sigma notations can be combined and added together. Now the pattern can be seen more clearly with the chart on the left. Each corresponding term added together will equal to n+1. Since there are n terms, n+1 is multiplied by n. In order to obtain the sum of all terms, n(n+1) is simply divided by 2, because n(n+1) is the sum of 2 expressions. In the end, the following expression is formulated:
n(n+1)2=n2+n2
In the end it can be inferred that
y=i=1ni=n2+n2
where n∈N. This statement is indeed equivalent to the equation found using the graphic calculator which proves the validity of this statement. One might also notice that triangular numbers are an additive resemblance of factorials, which represent the product of consecutive positive integers less than and equal to n.
The next task would consist of investigating the patterns of a stellar number. First up will be a 6-stellar, a stellar (star) shape with 6 vertices, meaning p=6. Before generating another general statement regarding 6-stellar number, the reader will be given insight to what a 6-stellar might look like:
Figure 3
Figure 3 consists of the first four 6-stellar numbers. For this investigation, the terms will be labeled as S1, S2, S3, and S4respectively thus a term of the sequence at any stage will be known as Sn. Similarly to the analysis of the triangular numbers, the 6-stellars will be analyzed for any potential patterns in effort to derive a general statement for the 6-stellar at stage Sn.
Table 3
Table 3 shows the tabulated data of the total number of dots, 1st differences, and the 2nd differences of 6-stellar numbers. Similar to the method used in triangular numbers, the data was collected by counting all the dots within each stellar and calculating the differences. After already discovering the general statement for the triangular numbers, the constant change in 1st differences or constant 2nd difference suggests that the general statement for 6-stellars will also be a quadratic function. Using the TI-84 Plus graphic calculator and following the same method mentioned in the task of triangular numbers, the following equation was given, y=6x2-6+1, where x∈N. Although the general statement is present, where do these numbers come from? After some thorough analysisof the relationship between 6-stellar numbers, an interesting observation was discovered.
Table 4
With the help of Table 4, where the different patterns have been organized, the pattern is much easier to visualize and interpret. The last 4 columns show the sum of the stellar numbers expressed in different forms and notations. The 1st column shows the sum in terms of adding the 1st difference to the previous term. The 2nd column replaces the previous term with a numerical value. The 3rd column takes that value and substitutes it with an expression of the sum of the previous term. After this process of logical reasoning, the pattern is now clearer to see. With exception to the first number, everything else can be factored out to a common factor of 12. By factoring everything out, the expression is recorded in the 4th column. At a first glance, nothing particular stands out, but the red bolded numbers correspond to the exact same numbers from the previous task of triangular numbers.
Table 5
Table 5 helps illustrate the pattern more clearly. With this set of data combined with the general statement regarding triangular numbers, the general statement for 6-stellars can be formulated by substituting corresponding variables. Let s=x because they will represent the same variable and by looking at the numbers from the table it can be inferred that n=s-1=x-1. For 6-stellars, the general statement would be in the form of y=1+Tn12, where Tn is the triangular number. Knowing that the general statement for triangular numbers is Tn=(n2+n2), and thatn can be substituted with x-1, everything can be combined into one andgive the following equation:
y=1+(x-1)2+(x-1)212
By expanding (x-1)2+(x-1) and multiplying it by 12, the equation simplifies to
y=1+(12x2-24x+12)+(12x-12)2=1+12x2-12x2=6x2-6x+1
Hence, it’s been derived that the general statement for a 6-stellar numbers is
where{n|n>0,n∈N}.y represents the total number of dots and n represents the term of the 6-stellar. This formula is equivalent to the equation found if one were to use the calculator to determine the quadratic function of the 6-stellars.
A valid theory cannot be formed with only one trial/sample, so therefore several more trials will be tested for other values of p. The explanation of the process of obtaining the general statement for the next p-stellars will be swift as the reader is now accustomed to the concept of triangular numbers and one concrete example of a stellar number.
Figure 4
A sample of a 4-stellar will be tested first. The image of what a 4-stellar looks like is given to the left in Figure 4. Notice, that a 4-stellar is not a square, a misinterpretation that is very commonlymade. After drawing all the diagrams using the computer and counting all the dots, the data was tabulated into Table 6 shown below.
Table 6
The data is organized into aneat combination of the tables made for the 6-stellar numbers. Once again, a constant 2nd difference means that the general statement will be a quadratic function and similar to the 6-stellar the total sum of Sninvolves the triangular numbers. As a matter of fact the relationship between Sn and n is exactly the same as it was for the 6-stellar, n=s-1=x-1. Now, by substituting the general statement for the triangular numbers into y=1+Tn8 this equation is formulated:
y=1+n2+n28=1+(x-1)2+(x-1)28
By expanding (x-1)2+(x-1) and multiplying it by 8, the equation simplifies to
y=1+(8x2-16x+8)+(8x-8)2=1+8-8x2=4x2-4x+1
Hence, it’s been derived that the general statement for a 4-stellar number is
where {n|n>0,n∈N}.y represents the total number of dots, and n represents the term of the 4-stellar. This formula is equivalent to the equation found if one were to use the calculator to determine the quadratic function of the 5-stellars with the given set of points from Table 6.
Going in numerical order, a 5-stellar is up next for analysis. The representation of a 5-stellar is given in Figure 5 to the right. Table 7 below shows the data obtained by counting the total number of dots and calculating the differences and the expressions for the sums.
Figure 5
Table 7
Yet again, the general statement will be a quadratic function due the constant 2nddifference and the expression of the total sum contains triangular numbers. It seems like a pattern is forming here, but no rushed conclusions will be made yet. The expression for 5-stellar is given by y=1+Tn10, where Tnis the general statement for the triangular numbers in which n=s-1=x-1. Therefore, this equation is formed:
y=1+(x-1)2+(x-1)210
By expanding (x-1)2+(x-1) and multiplying it by 10, the equation simplifies to
y=1+(10x2-20x+10)+(10x-10)2=1+10x2-10x2=5x2-5x+1
Hence, it’s been derived that the general statement for a 5-stellar number is
where {n|n>0,n∈N}.y represents the total number of dots, and n represents the term of the 5-stellar. This formula is equivalent to the equation found if one were to use the calculator to determine the quadratic function of the 4-stellars with the given set of points from Table 7.
Figure 6
A 6-stellar has already been analyzed in the beginning so a 7-stellar is next. The diagram for a 7-stellar is shown to the left in Figure 6. Parallel to the methods used in the previous 4, 5, and 6 stellars, the first 6 terms for the 7-stellar were drawn using a computer and all the dots were counted. The data was then tabulated into Table 8 below along with the expressions for the total sums of Sn.Table 8
It is evident that the general statement for a 7-stellar will also be a quadratic function because of the constant 2nd difference of 14. The expression for the stellar number still involves triangular numbers and is in the form of y=1+Tn14, where Tnis the general statement for the triangular numbers in which n=s-1=x-1. By substituting everything, this equation is formed:
y=1+(x-1)2+(x-1)214
By expanding (x-1)2+(x-1) and multiplying it by 14, the equation simplifies to
y=1+(14x2-28x+14)+(14x-14)2=1+14x2-14x2=7x2-7x+1
Hence, it’s been derived that the general statement for 5-stellar numbers is
where {n|n>0,n∈N}.y represents the total number of dots, and n represents the term of the 7-stellar. This formula is equivalent to the equation found if one were to use the calculator to determine the quadratic function of the 7-stellars with the given set of points from Table 8.
Table 9
With general statements formed for triangular numbers, 4-stellar, 5-stellar, 6-stellar, and 7-stellar, one final general statement will be concluded for a stellar with p vertices at stage Snin terms of n. Table 9 on the left puts all the general statements together into one perspective to assist with the processing of the general statement. All these statements are quadratic functions (y=ax2+bx+c) and they all appear to be of the same form. aand b are of the same numerical value and in each case of a stellar number, a and b correspond to p. From this observation, it can be predicted that the general statement is
where{p|p>0,p∈N} and {n|n>0,n∈N}. p is the number of vertices of a stellar and n is the term of the stellar.
Figure 7
To test the validity of this statement a simple substitution method will be used. The stellar number for a 4th term of a 5 vertices star will be y=542-5⋅4+1=61. Checking with the Table 7, the stellar number is indeed 61. The stellar number for a 3rd term of a 7 vertices star will be y=732-7⋅3+1=43. S3 for a 7-stellar is in fact equivalent to 43.The stellar number for a 5th term of a 6-stellar will be y=652-6⋅5+1=121, which is the correct value for S5of a 6-stellar. To further test the validity of this statement, different values for p and n will be used that have not been tested yet in this investigation. The stellar number for a 2ndterm of 8-stellar number will be y=822-8⋅2+1=17. After drawing the diagram for the 2nd term of 8-stellar (Figure 7), the total number of dots does equal to 17. From the tests so far, the general statement seems to valid one.Graph 1
Since this investigation did not have to deal with rational and irrational numbers, the degree of error is not a hefty one in this case. Possible mistakes could happen in the calculations or drawing the diagrams, but besides that there aren’t that many variables prone to significant error, so it’s safe to say that the general statement formulated is a fairly accurate one.This investigation mainly dealt with relatively small values of p and n, so this brings up a query of whether there are any limitations or not. The general statement is a quadratic function, so the limits of a quadratic equation will also apply to the limits of the general statement. The values for a, b, and c are insignificant because x2 is the dominant function. In a quadratic function, as x approaches infinity, y does not converge towards a particular value, so it does not have a limit, meaning the general statement will not be affected by big values ofn.This makes sense as the stellar number always increases with the following term, so logically there wouldn’t be any limit as the sequence progresses. Graph 1 above helps visualize the previous statements regarding limits. The graph was created with Microsoft Excel 2011. First the data from Table 8 of 7-stellar numbers was pasted onto the spreadsheet (only Sn and y were pasted). The table wasselected, and then a graph was created by following these steps: select the Insert tab and then choose the option shown in Figure 8, the 2-D Line Graph.On the graph it can be seen that as n increases, or in other words as it approaches infinity, the y value is always increasing showing no signs of a limit. Even though quadratic functions’ domain includes negative numbers, negative numbers and 0 are excluded from the domain, because it is impossible to have a negative term of a sequence and a 0 term does not exist.With regard tothe number of vertices,p>0 because it is not possible to have a stellar with a negative number of vertices or a stellar without any vertices.Figure 8
What this investigation has showed is that it is possible to derive a pattern out seemingly ordinary objects and shapes. At first no particular pattern can be seen, but after some thorough analysis one might notice something that ordinarily would be unnoticeable. The mathematical concepts of number sequences and quadratic functionsreally helped making this investigation a fluid task. Given more time, further investigation could be completed with regard to other possible relationships between triangular numbers and some other geometric shapes such as square numbers or perhaps thetotal number of lines in each stellar. The possibilities are endless and the model in this investigation can be applied to many other aspects of mathematics. Nonetheless, this investigation shows that with tedious analysis and effort, a pattern can be generated regarding even the most mundane objects in life provided there is consistency with the variables.
The diagram was created using GeoGebra. All the diagrams in this paper were created with GeoGebra.
by
yulin1994 (student)
