For example, in the first term, we see that there is one row only and in total, one dot in that row. In the next term, there are two rows, the first row still having ony one dot but the second row now having two dots. In the third term, ther are three rows, the first row same as the previous two terms, the second row with two dots and the third row with three dots, the same rule applies with the fourth and fifth term.
So now I have determined the pattern of this geometric shape, I apply the same law or rule to the next three terms.
The next three terms shown in the diagarm below:
As shown above, I added a sixth row for the sixth term, a seventh row for the seventh term and an eighth row for the eigth term eight, the total number of dots is the same with the row number.
In order to illustrate and list the patterns in a more organized way, I created a list, as shown below:
Part III: The general statement for triangular numbers
III.a Finding the general statement
Next, I need to generate a gerenal statement, the general statement that represents the nth triangular number in terms of n.
Since I am finding the nth term, then the equation would be the following shown below:
T(n) = 1 + 2 + 3 + 4 +...+ n (1)
Then, I wrote a conversed equation of (1), shown below:
T(n) = n + n-1 + n-2 + n-3 +…+1 (2)
The reason for creating a conversed equation, is because after adding both equations together, a common difference or ratio, namely a relationship between the terms. After combining (1) and (2) together:
T(n) = [1+n] + [2+(n-1)] + [3+(n-2)] + ... + [(n-1)+2] + [n+1] (3)
I get (3), and after simplifying (3), I got this:
T(n) + T(n) = (n+1) + (n+1) + (n+1) + … + (n+1) + (n+1)
T(n) + T(n) = n(n+1)
T(n) = (4)
Thus, I have successfully generated a general statement in the form of n.
III.b Testing the general statement
In this section, I have to test my general statement to insure that the statement fits the sequence just because of coincidence. I’m going to use my general statement to find the 9th and 10th triangular number.
Now I am going to find the 9th triangular number:
T(n) =
T(9) =
T(9) =
T(9) =
T(9) = 45
Now I am going to find the 10th triangular number:
T(n) =
T(10) =
T(10) =
T(10) =
T(10) = 55
To test my statement, I will need to present the actual triangular number, in form of a diagram:
From the diagram above, I counted that there are a total of 45 dots in the 9th term and a total of 55 in the 10th term. Thus, my general statement has been proven to be correct.
Part IV: Exploration of 6-stellar numbers
IV.a Investigation of 6-stellar numbers
In this section, I will investigate and familiarize myself with stellar numbers that has six vectors. Through this investigation, I hope that I will understand and grasp the concept of stellar numbers better.
The diagram below shows the first eight 6-stellar numbers:
As seen in the diagram, the simpler pattern that can be seen is that each term adds a larger star, larger in length. Furthermore, when closely reviewing the shapes, the difference becomes apparent. Each newly added star has 12 tips or points, the real difference lies in the line of the between the tips. The first star has non, the second star has one in between the joints, the third has the two, the fourth has three, and pattern continues from there.
From there, I used the information to change the equation from this one:
T(n) = (4)
to this one:
(5)
IV.b Testing validity of general statement
In this section, I will test the validity of my general statement, I will calculate the first eight terms to see if the outcome matches fro the diagram.
And thus, for the outcome, the answers match my previous findings from the diagram, so therefore, my general statement is valid.
Part V Exploration of 5-stellar numbers
V.a Investigation of 5-stellar numbers
In this section, I will investigate another stellar number group, in order to further understand the different stellar groups. To make my diagrams clear, I have enlarge the diagram to make them easier to see.
From the diagram above, I can work out the pattern of the 5 stellar numbers. From close inspection, I see that it is actually similar to 6 stellar numbers, except for the number of vectors. 6 stellar has 6 where as 5 stellar has 5. Apart from that, the pattern regarding the increase of dots is the same, where the number of dots between the lines increase by one, while the tips remain to be 10.
Therefore, from the information and the observation from the diagram, I can generate a general statement similar to (5):
(6)
V.b Testing validity of general statement
In this section, I will test the validity of my general statement to ensure that is works, I will do this by using the equation to calculate the first eught terms to see if it match my findings from the diagram.
Thus, from the calculations above I see that my general statement is valid, since all the outcome is the same with the numbers that I get from manually counting the number of dots from the diagram.
Part VI Universal general statement for stellar numbers
V.a Formation
In this section, I will need to form a universal general statement that could be applied to all stellar numbers. From the two previous sections, I find that that the general statement is tied to the number of vectors – p.
(7)
In the equation, p would equal to the number of vectors. For example, from the previous two sections, p would equal to 5 or 6. From that we can further simplify the statement to:
(8)
However, p cannot equal to negative numbers, 0,1,2 because vectors need more than 2 to become a shape, so it means that I will need to add limitation to the equation:
(9)
Thus, the universal general statement has been form, also with its limilations.
V.b Testing the general statement
In this section, I will test the validity of the general statement to insure that it works. I will do this by randomly insert numbers in the places of p and n. Then I will attch diagram and manually compare both results or total number of dots to see if the numbers match.
First I will start with the 4th term of the 4-stellar number:
Then, I will check with a diagram:
Next I will test it with the 5th term of 10-stellar number:
Then I will attach a figure:
Figure 7: 5th term 10-stellar number
Thus, from the proof and testing above, it shows that my general statement can accurately find the total number of dots in the different terms of only vectors of stellar number, provided that it falls between the limitation.
Part VII Conclusion
In this internal assessment, my ultimate goal was to find the general statement for the stellar numbers. I have succeeded in this area. My general statement was able to find the total number of dots and find the answer correctly. However there are certain limitations in my statement, for example, it could only be used to find shapes, not lines. Meaning that, the subject or term I want to find must be a shape, it must have a plane in order for the equation to work. A shape with only to vectors, that is a line or one vector, a dot cannot be calculated. Also, p can’t be a negative number, since no shape can have a negative number.
I think that my general statement is the best way to calculate the terms, since it is easy to use and it is a lot quicker. Counting manually will be very time-consuming and a lot if the vectors is very large, and the term is very big there is a chance that I will count it wrong, since it is very easy to overlook some of the dots, and if you think that you missed something, then you have to start all over again. So it is better to use my general statement.
During the investigation, I used different kinds of technology to help me, including paint for the diagrams, math type for the equations. It is important to use different software to help because it can help show how technology can be part of real life; these programs can help solve mathematical problems. These program help speed up my investigation, and I will definitely use them again.
