2009/025) Let t_n denote the nth triangular number. For what values of n does t_n divide the sum t_1+t_2+...+t_n?

Well we know t_m = (m²+m)/2

We want S_n = Σ_n=1_m [ t_m ] = Σ_n=1_m [ (m²+m)/2 ]
Applying Faulhaber's Formulas,
S_n = [ (2n³+3n²+n)/6 + (n²+n)/2 ] /2
= (2n³+3n²+n + 3n²+3n) /12
= (2n³+6n²+4n) /12
= n(n+1)(n+2) /6

So the question is for what values of n does
n(n+1)/2 | n(n+1)(n+2)/6

RHS/LHS = (n+2)/3 and LHS is a factor if and only if this is an integer

so n + 2 = 0 mod 3 or n = 1 mod 3

so for n = 1 mod 3 the value t_n devides the sum