Question

explaining if a sequence has direct variation
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.
the direct variation is

Explanation:

Step1: Recall direct - variation formula

For direct variation, the relationship is $y = kx$, where $k$ is a constant. Let $y$ be the triangular number and $x$ be its position in the sequence.

Step2: Calculate ratios for each pair

For $x = 1,y = 1$, the ratio $\frac{y}{x}=\frac{1}{1}=1$.
For $x = 2,y = 3$, the ratio $\frac{y}{x}=\frac{3}{2}=1.5$.
For $x = 3,y = 6$, the ratio $\frac{y}{x}=\frac{6}{3}=2$.
For $x = 4,y = 10$, the ratio $\frac{y}{x}=\frac{10}{4}=2.5$.
For $x = 5,y = 15$, the ratio $\frac{y}{x}=\frac{15}{5}=3$.

Step3: Check if the constant of variation exists

Since the ratios $\frac{y}{x}$ are not the same for all pairs of triangular numbers and their positions in the sequence, there is no constant $k$ such that $y = kx$.

Answer:

No, there is no direct variation between a triangular number and its position in the sequence because the ratios of the triangular number to its position are not constant.