proving partial sums are arithmetic serie

Given $3$ arithmetic progressions with same difference $d$:

$(1)$ $a_1,a_2,a_3,...$ $(2)$ $b_1,b_2,b_3,...$ $(3)$ $c_1,c_2,c_3,...$.

defined: $S_n=a_1+a_2+a_3+...+a_n$

$T_n=b_1+b_2+b_3+...+b_n$

$R_n=c_1+c_2+c_3+...+c_n$

Need to prove that if $a_1,b_1,c_1$ is an arithmetic progression so $S_n,T_n,R_n$ is also an arithmetic progression.

I tried to use the fact that $2b_1=a_1+c_1$ and $a_2-a_1=b_2-b_1=c_2-c_1=d$ but i'm stuck

Any idea?

Thanks.

EDIT - $a_n,b_n,c_n$ are arithmetic progressions