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.

Answer 1

Hello,

$T_{1} =1\\ T_{2} =3=1+2\\ T_{3} =6=1+2+3\\ T_{4} =10=1+2+3+4\\ T_{5} =15=1+2+3+4+5\\ ...\\ T_{n} =1+2+3+....+n= \dfrac{n*(n+1)}{2}$

Answer 2

Answer: 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 so on, do not lie on a line.