Sl Math Internal Assessment: Stellar Numbers

SL Math Internal Assessment: Stellar Numbers

374603
Mr. T. Persaud
Due Date: March 07, 2011
Part 1: Below is a series of triangle patterned sets of dots. The numbers of dots in each diagram are examples of triangular numbers.
Let the variable ‘n’ represent the term number in the sequence. n=1 n=2 n=3 n=4 n=5

1 3 6 10 15 n=6 n=7 n=8

21 28 36 Term Number (n) | Number of Dots (tn) | First Difference | Second Difference | 1 | 1 | - | - | 2 | 3 | 2 | - | 3 | 6 | 3 | 1 | 4 | 10 | 4 | 1 | 5 | 15 | 5 | 1 | 6 | 21 | 6 | 1 | 7 | 28 | 7 | 1 | 8 | 36 | 8 | 1 |
As we can see from the chart above, there is a growing increase in the differences between each consecutive set of numbers of dots. The difference between the number of dots in sets 2 and 1 is 2, the difference between the number of dots in sets 3 and 2 is 3, the difference between the number of dots in sets 4 and 3 is 4, and so on. The differences between each consecutive pair of triangular numbers increase constantly by 1 each time. This dictates that there is a constant second difference of 1 for this sequence.
The term values, or in this case the number of dots, increase neither arithmetically nor geometrically. This piece of information, shows that the general statement will require an exponent on the variable ‘n’ in order to define a relationship between the term number and the term value. The constant second differences indicate that ‘n’ will be raised to the