triangular numbers can be represented with equilateral triangles formed by dots. the first five triangular…

Explanation:

Step1: Recall direct - variation formula

Direct variation is of the form $y = kx$, where $k$ is a constant. Let $y$ be the triangular number and $x$ be its position in the sequence.

Step2: Check for a constant ratio

For the first triangular number ($x = 1,y = 1$), the ratio $\frac{y}{x}=\frac{1}{1}=1$. For the second triangular number ($x = 2,y = 3$), the ratio $\frac{y}{x}=\frac{3}{2}=1.5$. For the third triangular number ($x = 3,y = 6$), the ratio $\frac{y}{x}=\frac{6}{3}=2$. For the fourth triangular number ($x = 4,y = 10$), the ratio $\frac{y}{x}=\frac{10}{4}=2.5$. For the fifth triangular number ($x = 5,y = 15$), the ratio $\frac{y}{x}=\frac{15}{5}=3$.

Step3: Analyze the results

Since the ratios $\frac{y}{x}$ are not constant for different values of $x$ and $y$, there is no direct variation.

Answer:

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