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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Sean Okundaye


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.

   

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.

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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 ...

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