# Stellar Numbers Ia

Topics: Triangular number, Polygonal number, Number Pages: 8 (1742 words) Published: March 14, 2013
Patterns and sequences are the basis of mathematical understanding. Based on geometry alone, many special patterns evolve, such as the square numbers, triangular numbers, and much more. The Stellar Numbers are mostly used in astronomy and astrology. Stellar Numbers are figurate numbers based on the number of dots that can fit into a midpoint to form a star shape. The points of the star determine the number of points plotted around the midpoint. Triangular numbers is a figurate number system that can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one. The triangular numbers from that pattern are 1 followed by 1+2 followed by 1+2+3 and so on. From the pattern of the triangular numbers, this infinite serious starts with 1, 3, 6, 10, 15… With this pattern, calculated by counting, the next three terms would be 21, 28, and 36. To derive a formula from this pattern, we can see that x is repeated and the number goes up each time by one. After using the rules of the sequences and a few checks, the final formula results inx(x+1)2 where x is any natural number. I found this formula with the calculator with steps show below. To prove this formula, there is the typical guess and check formula where the number of dots in the next triangle is counted for. Strangely when noticed clearly, the triangular numbers can be found in the third diagonal of Pascal’s triangle, starting at row 3 as shown in the diagram. The first triangular number is 1, the second is 3, the third is 6, the forth is 10, and so on. This following drawing below shows the triangular pattern of evenly space dots. In this first picture, the triangular numbers with three more terms is completed.

Solution One: Technology
The next step is to find a general statement that represents the nth number of dots in the triangular number series in terms of x. Let X represent the stage and let Y represent the corresponding triangular number: x| 1| 2| 3| 4| 5| 6| 7| 8|

y| 1| 3| 6| 10| 15| 21| 28| 36|

The calculator used in the production of this formula was Texas Instrument TI-84 Plus Silver Edition. Select STAT. Punch in the value of X in list 1 and the values of Y in list 2. Select CALC 5 to Quadratic Regression to get y=ax2+bx+c where a, b=0.5 and c=0. y=0.5x2+0.5x

=0.5(x2+x)
=x(x+1)2 General Statement

Solution Two: Caveman
To derive a formula in which X and Y are related to find the next terms, it will either be a geometric sequence, arithmetic sequence, or neither. In this case, it is neither arithmetic sequence nor a geometric sequence because the ratios of each term in Y do not equal and the values of differences are not the same. So there has to be another way to derive the formula. From calculations, differences between two adjacent terms have a clear pattern/arithmetic sequence.

x| 1| 2| 3| 4| 5| 6| 7| 8|
y| 1| 3| 6| 10| 15| 21| 28| 36|
Differences:
234 5 6 7 8

If the left side of all the calculations are added together, all the terms except Yx canceled because of the positive and negatives. The right side of the calculations adds up is the sum of the arithmetic sequence from 1 to x. In this manner, we can easily calculate the relation between Y and X. (Details following)

Task: to find Y in terms of X.
Y1 =1
Y2-Y1=2
Y3-Y2=3
Y4-Y3=4

Yx-1-Yx-2=x-1
Yx-Yx-1=x
Y1 + Y2-Y1 + Y3-Y2 + Y4-Y3 …. + Yx-1-Yx-2 + Yx-Yx-1 = 1 + 2 +3 +4 + … +x If the left side of all the calculations are added together, all the terms except Yx canceled because of the positive and negatives. The right side of the calculations adds up is the sum of the arithmetic sequence from 1 to x. In this manner, we can easily calculate the relation between Y and X.

Yx = 1+2+3+ 4 + … + x
= x(Y1+Yx)2
= x(x+1)2 General Statement...