# Stellar Numbers Ia

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...

Please join StudyMode to read the full document