Question

Sample Response: No, the triangular numbers are not a direct variation. There is not a constant of variation between a number and its position in the sequence. The ratios of the numbers to their positions are not equal. Also, the points (1,1), (2,3), (3,6) , and 87 so on, do not lie on a line.

What did you include in your response?
Check all that apply.
a. The sequence is not a direct variation.
b. There is no constant of variation.
c. The sequence cannot be described by the equation y=kx.
d. The points do not lie on a line.

Answer 1

No, the triangular numbers do not represent a direct variation because they do not hold a constant ratio, are not linear, and their plot points do not align on a straight line but instead form a parabolic curve.

Step-by-step explanation:

The triangular numbers are a sequence where each number represents a sum that could form an equilateral triangle. To assess whether the triangular numbers represent a direct variation, we can explore three aspects:

Consistent ratios (constant of variation)

Representation by a linear equation

Alignment of points on a straight line when plotted

The sequence of triangular numbers does not have a constant ratio between successive terms and their positions. Triangular numbers are given by the formula Tn = n(n+1)/2, which is quadratic, not linear like direct variation expressed by y=kx. Additionally, when plotted on a graph, the points that represent the triangular numbers do not lie on a straight line but rather on a parabolic curve. Therefore:

The sequence is not a direct variation.

There is no constant of variation.

The sequence cannot be described by a linear equation y=kx.

The points do not lie on a straight line.