Question

triangular numbers can be represented with equilateral triangles formed by dots. the first five triangular numbers are 1, 3, 6, 10, and 15. is there a direct variation between a triangular number and its position in the sequence? explain your reasoning.

Explanation:

Step1: Recall the definition of direct variation

For two variables \(x\) (position) and \(y\) (triangular number), if \(y = kx\) (where \(k\) is a constant), then there is a direct variation.

Step2: Check the ratios for the given data

Let \(x_1 = 1,y_1=1\), ratio \(r_1=\frac{y_1}{x_1}=\frac{1}{1} = 1\).
Let \(x_2 = 2,y_2 = 3\), ratio \(r_2=\frac{y_2}{x_2}=\frac{3}{2}=1.5\).
Let \(x_3=3,y_3 = 6\), ratio \(r_3=\frac{y_3}{x_3}=\frac{6}{3}=2\).

Since the ratios \(\frac{y}{x}\) are not constant (\(1
eq1.5
eq2\))

Answer:

No, there is no direct variation. The ratios of triangular numbers to their positions are not constant.