Stellar numbers are sequence of numbers that follow a certain pattern, when we plot it into a diagram it will create a geometrical star-like shape. Each stellar number has its own vertices and number of dots, the formula will be different for every vertices. We are now going to determine the number of dots in each stage, and then we could see the pattern of the number of dots. Finally, we can generate a simplified formula or general statement for to find the number of dots in each stage.
Triangular numbers are sequence of number that has certain pattern and if we plot it into a diagram it will create a geometrical triangular shape. Each triangular number has its own number of dots and formula. Now, we will start with triangular number first. Aim: In this task I will consider geometric shapes which lead to special numbers. The simplest example of these are square numbers, 1,4,9,16 which can be represented by squares of side 1,2,3 and 4. The following diagrams show a triangular pattern of evenly spaced dots.
This can be explained, because stellar numbers must at least have 1 vertice. Explain how you arrived at the general statement To find the general statement, We must find the pattern of the nth term of the stellar number, by trying to find the formula of every stellar number with different vertices. I found that the general statement is in the form of quadratic equation with “1” as the value of c. The coefficient A and B have the same, which correspond to the number of vertices in each stellar number. After that, I need to test the general statement for the other stellar number to prove its validity. Conclusion The general statement of triangular number is Sn = 1/2n2 + 1/2n.
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I generated it by identifying the sequence of triangular number, and then create a formula that follows the pattern and it is proven to be correct because it can be used to find the number of dots in any term except the term ? 1. For the stellar numbers, I have found the general statement that works for various stellar numbers with different vertices, which is Sn= pn2-pn+1. Yet, there is a limitation of this general statement, it cannot be used when p > 1. To find the general statement, I have identified the pattern of the sequence, by trying to find the number of dots in each term. P in this formula represents the number of vertices in stellar numbers, n here, corresponds to the term of stellar numbers.
