Given the fact that our current sequence is 1, 3, 6, 10, 15… we need to find the nth term.
Looking at the sequence, we see that there difference between each term increases by 1 so for example the difference between u1 and u2 is 2 and the difference between u2 and u3 is 3. Therefore given our current sequence, we can say that:
This can be simplified into the following equations:
We need to find the next three terms and since we already have terms u1 – u5, the terms we need to find are u6, u7 and u8. To find these terms, we need to input 6, 7 and 8 into the above equation. Therefore:
Now that we have found these terms, we can display them in the form of a triangular pattern which can be seen on the next page.
If we now turn our attention to stellar (star) shapes, these have p vertices which then lead onto “p-stellar numbers”. If you look at the first four examples below, we can see stars with six vertices shown in stages s1 – s4.
You can see that at each stage there are six vertices on the shape and so we can say that each shape has a “6-stellar number”. This is the total number of dots in the above diagram and there is a way in which we can find the number of dots.
TASK 2: Find the number of dots (i.e. the stellar number) in each stage up to s6. Organise the data so that you can recognise and describe any patterns.
Having been presented with the first four stages, I have been able to count the number of dots manually and have produced a sequence consisting of 1, 13, 37 and 73 dots from stages s1 to s4 respectively. The difference between each stage was 12, 24 and 36 respectively and the difference between each of these stages was 12. Therefore, we can find out the next two terms by continuing the sequence of adding 12 to the last stage. From there we can add the value to the last stage of the second value and complete our sequence up to s6. This can be seen below.
1 13 37 73 121 181
12 24 36 48 60
12 12 12 12
Therefore can now produce two stellar shapes for s5 and s6 which can be seen on the next page.
TASK 3: Find an expression for the 6-stellar number at stage s7. Find a general statement for the 6-stellar number at stage sn in terms of n.
Since we had to find the difference twice, the formula for the sequence is a quadratic equation, i.e.:
Therefore, an expression for the 6-stellar number at stage s7 would be:
To be able to find a general statement for the 6-stellar number at stage sn, we need to work out the value of a, b, and c. Using the first three terms as examples, we can say that:
We find the values of a, b, and c. We can start by subtracting s2 from s1:
We need to also subtract s3 from s2:
With the values we have obtained, we can now find out the value of a:
Now that we have found the value of a, one can find the value of b:
Since we now know the values of a and b, we can substitute these values into s1:
1
Now that we now know all the values, we can now form the equation of the sequence which is as follows:
TASK 4: Repeat the previous tasks for other values of p. 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.
If one was to use a stellar shape with 5 vertices as an example, the 5-steller number for stages s1 to s4 would be as follows:
1 11 31 61
10 20 30
10 10
This can be seen in the stellar shapes below that are produced with a 5-stellar number.
Like before, we will need to solve a quadratic equation to find a general statement for the stellar number at sn, this time a 5-stellar number. We can use the first three stages:
To find the values of a, b, and c, we can subtract s2 from s1:
We need to also subtract s3 from s2:
With the values we have obtained, we can now find out the value of a:
Now that we have found the value of a, one can find the value of b:
Since we now know the values of a and b, we can substitute these values into s1:
1
Therefore, we can say that with a 5-stellar number:
Now considering a stellar shape with 7 vertices and so a 7-stellar number, stages s1 to s4 are as follows:
1 15 43 6
14 28 42
14 14
This can be seen below in the form of stellar shapes:
Once again we will need to solve a quadratic equation to find a general statement for the stellar number at sn, this time a 7-stellar number. We will use the first three stages:
To find the values of a, b, and c, we can subtract s2 from s1:
We need to also subtract s3 from s2:
With the values we have obtained, we can now find out the value of a:
Now that we have found the value of a, one can find the value of b:
Since we now know the values of a and b, we can substitute these values into s1:
1
Therefore, we can say that with a 7-stellar number:
Looking at the general statements that have been produced, one can see that there is a trend. For each of the stellar numbers I have tested, the p value has been equal to the value of both a and b. Therefore, I have come to the conclusion that:
TASK 5: Test the validity of the general statement and discuss the scope or limitations of the general statement.
When testing, the validity I found that when I made the p value 2, a problem occurred:
1 5 13 27
4 8 14
4 6
2
As can be seen from the above sequence, there are four levels. Therefore, the equation of the sequence should be cubic as opposed to quadratic and so this goes against the general statement. If the p value was 3, the same problem did not occur:
1 7 19 37
6 12 18
6 6
Therefore, one should probably also conclude that:
I.e. p cannot be less than 3.
The limitation of my investigation is the fact that I did not test every single stellar number and therefore cannot be certain that every number that is not less than 3 will work. Furthermore, when working with values of n I only went up to as high as 4 and so I cannot be certain that my general statement is correct for numbers such as 80 or 200.