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

No. "Direct variation" means T_n = k·n for some constant k (the ratio T_n/n must be constant). For triangular numbers

T_n = 1 + 2 + … + n = n(n + 1)/2,

so T_n/n = (n + 1)/2, which depends on n (1, 3/2, 2, 5/2, 3, …). Because the ratio is not constant (and T_n is quadratic in n, not linear), triangular numbers do not vary directly with their position.