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

direct variation is y = kx where y would be the value of the number and x would be its position in the sequence, k is a constant

so for consecutive values y/x would be a constant k

In this case its not true becuse for example 3/1 = 3 and 6/3 = 2

so there is no direct variation here.

Answer 2

position in the sequence triangular number relation

1 1 1 = 1*(1+1)/2

2 3 3 = 2(2+1)/2

3 6 6 = 3(3+1)/2

4 10 10 = 4(4+1)/2

5 15 15= 5(5+1)/2

Call n the position in the sequence, then the triangular number is: n(n+1)/2