Math SL Investigation Type 2 Stellar Numbers This is an investigation about stellar numbers, it involves geometric shapes which form special number patterns. The simplest of these is that of the square numbers (1, 4, 9, 16, 25 etc…) The diagram below shows the stellar triangular numbers until the 6th triangle. The next three numbers after T5 would be: 21, 28, and 36. A general statement for nth triangular numbers in terms of n is: The 6-stellar star, where there are 6 vertices, has its first four shapes shown below:
The number of dots until stage S6: 1, 13, 37, 73, 121, 181 Number of dots at stage 7: 253 Expression for number of dots at stage 7: Since the general trend is adding the next multiple of 12 (12, 24, 36, 48 etc…) for each of the stars, so for S2 it would be 1+12=13, and for S3 it would be 13+24=37 General statement for 6-stellar star number at stage Sn in terms of n: For P=9: Since S1 must equal 1 then we can prove this formula by showing that:
So the first six terms are: 1, 19, 55, 109, 181, 271 Therefore the equation for the 9-Stellar star at For P=5: Since S1 must equal 1 then we can prove this formula by showing that: So the first six terms are: 1, 11, 31, 61, 101, 151 So the expression for 5-Stellar at General Statement for P-Stellar numbers at stage Sn in terms of P and = For P-Stellar number equals 10: For P-Stellar number equals 20: The General Statement works for all number fro 1 to positive infinity.
The equation was arrived at since the sum of arithmetic series can be found using , since the difference is always 2P then we can replace 2P with d, and since u1 is always equal to 1, we can replace it with 1 every time. The at the end of the equation serves the purpose of making the difference between the numbers in the series constant. This form of the equation will allow for only one variable to change, which will be . One of the things the student realized while solving this investigation was that the second term is always equal to , but the derived equation which is also works.