In the table on the left, I observe that from the y value 1 to the y value 3 there is an increase of 2. From the y values 3 to 6, there is an increase of 3. From the y values 6 to 10, there is an increase of 4. This shows that it goes: +2, +3, +4, which is then increasing by 1 between each of those numbers.
This follows the form of an arithmetic sequence, because each new term is equal to the sum of the previous term plus the term number. A formula for this sequence can then be derived form the formula of an arithmetic sequence, shown below.
n represents the term number, a represents the number of dots in the first term, and d represents the common difference between the terms.
Since the first team has one dot, this means that for this sequence, a=1. The difference between each term is represented by d. Therefore, d=1.
By substituting these values into the arithmetic sequence formula, I get the equation:
Which is then simplified to:
This formula above can be used to calculate the number of dots of any triangular number is the sequence, n being greater than 0.
Graphing the points from the table above, gives me this graph:
The equation:
is correct because the points in the table fall exactly on the graphed line.
Above, the equation: also follows the general equation y=ax2+bx+c.
If I move around the equation to fit the general form, .
Therefore, a= ½, b= ½, and c=0.
Next, I will be considering stellar shapes with p vertices, leading to p-stellar numbers. I will be showing you stages S1-S6 in diagrams below.
From stage 1 to 2, the amount of dots increases by 12. From stage 2 to 3, the amount of dots increases by 24. From stage 3 to 4, the amount of dots increases by 36. This pattern continues throughout the stages by going +48 and then +60. Looking at the numbers 12, 24, 36, 48, and 60 you can see that there is an increase of 12 between each number. This shows that on each new layer, there are 12 new dots compared to the previous layer. This means that if the term number is represented by n, then the following formula can be produced:
Sn=12(n-1) + S(n-1) where n>1
By using this formula, the number of dots in any 6-stellar term can be found.
For example, S6 can be found by:
S6=12(6-1)+S(6-1)
This is then simplified into: S6=72-12+121 and S6= 181
This formula, however, requires the number of dots in the previous term of work. Therefore, a better formula must be devised.
Ignoring the first term, the rest of the data seems to follow the trend of an arithmetic sequence, similar to the triangular example I first showed you.
Using 12 as the first term and adding one to the equation can create a formula for the 6-stellar shapes.
This equation can be used to calculate the number of dots in S7. Because the first term is 12 (which is actually the value of S2), n will actually have the value of one less than 7. This gives the equation:
After simplifying this, S7= 253.
This proves the pattern and is shown in the table below.
Using the same equation that I used to calculate S7, a general formula for Sn can now be derived.
This value of a applies to S2 not S1, and therefore the formula will be equal to Sn+1.
Sn+1 =
After fully simplifying this equation, I get:
Since this equation is equal to Sn+1, I am going to substitute n= (n-1) into the equation to find the value of Sn.
Simplifying to:
Graphing this equation (y=6x2-6x+1) results in:
I plotted the points from this table on the right and the points lined up on the parabola perfectly, which therefore verified that the equation was correct. In addition, it shows that all examples so far have resulted in a similar shaped parabola graph which illustrates how these examples are all similar ideas resulting from the equation y=ax2-bx+c.
This general equation can now be used to find the number of dots in 6-stellar shapes. For any p-stellar number, the following formula can be used to find the number of dots by substituting p into the place of “6” for the 6-stellar shape equation.
However, there are some limitations for p. There can be no values less than zero, because there cannot be negative number of dots in the shape. This means p must be greater than or equal to zero. However, p cannot be equal to zero either because this would result in term being equal to one each term, which would not be a stellar shape. Therefore, p>0. In addition, p cannot equal 1 because this would result in an impossible shape. So because of these limitations, p must be greater than 1.
The general statement for p-stellar shapes is:
Sn= pn2-pn+1, p>1
