or
Lets check this general statement with the following table:
One can confirm from this table that the general formula of the nth triangular number in terms of n provides exactly the same results as the consecutive formula from Task 1. Finally, one can confirm that the two formulas are identical by the following calculation:
Substituting with the general statement:
and (n-1)
Opening the brackets, one can see that the two sides are identical; therefore the general formula for n is the same as the consecutive formula from Task 1.
Thus one can conclude that the general statement for the triangular numbers is as following:
Task 3
For the stellar shape with six vertices, find the number of dots (stellar number) in each stage up to S6. Organize the data so that you can recognize and describe any patterns.
The first four representations for a star with six vertices (p=6) are shown in the four stages S1 – S4 above. From visual observations, one can notice, that each subsequent figure derives from the previous one by adding an outer star of a similar shape (with the same number of vertices). Thus, each figure at stage n consists of an n number of stars, placed inside one another similar to “Matryoshka dolls”. Consider the features of the consecutive stars. Each consecutive star is similar to the previous one with one notable exception: there are two more additional dots in each of the vertices.
This diagram demonstrates the new dots added to the outer simple star at each consecutive stage (from S2 to S3 in this particular case). Using this visual representation, we can model the visual pattern of the stellar shapes for the stages S5 and S6.
To find the number of dots in each stage, one can simplify each shape as a combination of a number of simple stars with 6 vertices and the dot in the middle. For example: the shape S3 consists of two simple stars and one dot in the middle; S4 consists of three simple stars and the dot in the middle, etc. The total number of dots in any particular shape is the sum of the dots of comprising it simple stars, plus one dot.
Lets denote the number of dots of the outer simple star of the shape Sn as Fn (F1 = 0).
For example: F2 is to the number of dots in the outer star of S2 and equals 12. The total number of dots is
F2 + 1 = 12
We can also observe a simple sequence of the Fn. From the “Diagram for Increased Number of Dots” above, one can derive that:
Fn= Fn-1 +2*6
Thus, each consecutive outer simple star has 12 more dots than the previous one (two per each of the six vertices). We can summarize all these findings in the following table:
Lets denote by Tn the stellar number for the stage Sn. Judging from the table 3.1, we have the following result of the calculations:
Task 4
Find an expression for the 6-stellar number at stage S7.
Using the table 3.1 from the previous task, we can derive the number of dots in the outer simple star (F7) at the stage S7.
Thus, F7 equals to 72 dots.
Using the formula from the table 3.1 in the previous task, we can derive an expression for the 6-stellar number at stage 7:
T7 =
Where Tn is the stellar number (number of dots) and Fn is the number of dots in the respective simple stars.
From the table 3.1, we know the values from to calculate T7:
T7 =
Thus, the number of dots in the 6-stellar number for the stage S7 equals 253
Task 5
Find a general statement for the 6-stellar number at stage Sn in terms of n.
In order to find the general statement in terms of n, we have to generalize the formula for Tn – the stellar number.
To begin with, lets re-work the table 3.1 from Task 3.
Observing the sequence from the last column, we can continue it to the stage S7, for example:
or to any subsequent stage S7:
We will now attempt to simplify this formula. Lets temporarily return to Task 1. As one can notice, the sum of 1+2+3+4+…+(n-1) equals to the number of dots in the triangular pattern at the stage (n-1), or to the (n-1)th triangular number Gn-1 . One can visually confirm this observation if he starts counting the dots of the triangular pattern diagram from the lower left corner and counts them layer by layer up to as depicted in the next diagram: 1+2+3+4+5…
We have already found the general statement for the nth triangular number Gn in terms of n:
If we look at the Task 2, we can now use this formula to calculate Tn:
Where, represents (n-1)th triangular number Gn-1. Opening the brackets, we find the general statement for the 6-stellar number at stage Sn in terms of n:
Let us verify this formula in the table below:
A shown by the table, the two columns provide identical results thus supporting the validity of the general statement:
Where Tn is the stellar number (total number of dots in the stellar shape at the stage Sn).
Task 5
Repeat the steps above for other values of p (number of vertices).
Lets consider the diagram for the increased number of dots from Task 3:
We can draw similar diagrams for stellar shapes with three or four vertices.
These diagrams demonstrate the new dots added to the outer simple star at each consecutive stage (from S2 to S3 in this particular case). Using this visual representation, we can model the logical pattern of the stellar shapes with three and four vertices: at each consecutive stage, the new outer simple star has two more dots per each vertices comparing to its predecessor. We can summarize this in the following table:
Using the equivalent of the table 3.1, we can calculate the total stellar numbers for p=3 and p=4.
We can now see that there are two common denominators in the two columns - 6 in case p=3 and 8 in case p=4:
Using the analogous formula from Task 5 () and substituting 12 for 6 and 8 respectively, we can solve for the stellar numbers in case of p=3:
And for p=4:
Task 6
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
By comparing the general statements for the p-stellar numbers:
p=3 p=4 p=6
We can conclude that the changing variable in these equations is p, so the final general statement in terms of p and n for all p-stellar numbers of any value of p at stage Sn is:
or
Where p is the number of vertices. Henceforth, we will have to test this general statement.
Task 7
Test the validity of the general statement.
Lets consider the stellar shapes with 5 vertices (p=5). We can calculate the number of dots (the stellar number) manually. Here they are:
We can now verify the general statement in the following table:
As one can see, the numbers are identical, which supports the validity of the formula.
Another example to prove the general statement as a stellar with 7 vertices (p=7) is going to be attempted below:
Again, as one can notice, the numbers are identical, proving the general statement’s authenticity.
Task 8
Discuss the scope and limitations of the general statement.
We have tested the general statement Tn = for a sufficient range of stellar structure with number of vertices from 3 to 7. Building the stellar structures with larger number of vertices is quite time consuming and thus further checks have to be done by a sophisticated computer program. However, the logic behind this general statement remains the same for any number of vertices, which leads us to the conclusion that the formula is correct. One has to use it properly, only for the values of n higher than 0 and for the values of p higher than 2, as one cannot imagine stellar shapes with one or two vertices. Technically it is possible to use any, even negative numbers, but we find no logical context for such calculations.
Task 9
Explain how you arrived at the general statement.
There were three key findings that helped me to arrive at the general statement. The first realization was that each new outer star has two more additional dots per each vertex. This allowed me to calculate the number of dots in any outer star using the recurrent formula.
Secondly, I noticed that each formula of the stellar number consists of one inalienable part: the sum of (0+1+2+3…+(n-1)).
Finally I noticed that this part above is in fact the triangular number from the Task 2.
Putting these three observations together, I found the formula, which was later confirmed by a decent number of verifications.
