First, the correct spelling is "independ[b]e[b]nt". Now, if [X1, X1 + X2, X1 +X2 + X3,....,X1 +X2.....XK] were not independent the there would exist numbers a1, a2, ..., aK, not all 0, such that a1X1+ a2(X1+X2)+ a3(X1+ X2+ X3)+ ...+ aK(X1+ X2+ ...+XK)= 0. rewrite that to collect "like" Xs and show that would imply that [X1, X2, ..., X] are not independent.
2. If [X1, X2, X3,......XK] is independant, show that [X1, X1 + X2, X1 +X2 + X3,....,X1 +X2.....XK] is also independant.
Same as 2.
3.If [Y, X1, X2, X3,.....XK] is independant show that [Y + X1, Y + X2, Y + X3,....Y+XK] is also independant