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Stellar numbers and triangular numbers. 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.

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Introduction

Sean Okundaye

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By | Sean Okundaye

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standard level mathematics type i coursework

Stellar Numbers


Contents

  • Aim of the investigation………..page 3
  • TASK 1: Complete the triangular number sequence with three more terms. Find a general statement that represents the nth triangular number in terms of n………… page 3
  • 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………… page 4
  • 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………… page 5
  • 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………… page 7
  • TASK 5:Test the validity of the general statement and discuss the scope or limitations of the general statement………… page 9

Aim of the investigation: In this task I will consider geometric shapes which lead to special numbers. The simplest examples of these are square numbers, 1, 4, 9, 16, which can be represented by squares of side 1, 2, 3 and 4.

Triangular numbers (i.e. 1, 3, 6, 10…) are numbers that can be displayed in the form of a triangular pattern composed of evenly spaced dots. This can be seen below.

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...read more.

Middle

We find the values of a, b, and c. We can start by subtracting s2 from s1:

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We need to also subtract s3 from s2:

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With the values we have obtained, we can now find out the value of a:

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Now that we have found the value of a, one can find the value of b:

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Since we now know the values of a and b, we can substitute these values into s1:

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Now that we now know all the values, we can now form the equation of the sequence which is as follows:

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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.image05.png

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:

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To find the values of a, b, and c, we can subtract s2 from s1:

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...read more.

Conclusion

a, one can find the value of b:

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Since we now know the values of a and b, we can substitute these values into s1:

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Therefore, we can say that with a 7-stellar number:

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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:

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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:

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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:

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1          7          19          37

6          12          18

6          6

Therefore, one should probably also conclude that:


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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.

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...read more.

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