E.697. Arătați că numărul S=63+133+203+…+(7n−1)3+15n ⋮ 7, ∀n∈N.S=6^3+13^3+20^3+ \ldots + (7n-1)^3 + 15n ~\vdots~ 7,~\forall n \in \N.S=63+133+203+…+(7n−1)3+15n ⋮ 7, ∀n∈N.
S=(7−1)3+(14−1)3+(21−1)3+…+(7n−1)3+(14n+n)=S=(7-1)^3 + (14-1)^3 + (21-1)^3 + \ldots + (7n-1)^3 + (14n+n)=S=(7−1)3+(14−1)3+(21−1)3+…+(7n−1)3+(14n+n)= =(M7−1)+(M7−1)+…+(M7−1)⏟n paranteze+M7+n==\underbrace{(M_7-1)+(M_7-1)+\ldots+(M_7-1)}_{\text{n paranteze}} + M_7+n==n paranteze(M7−1)+(M7−1)+…+(M7−1)+M7+n= =n⋅M7−n+M7+n=M7.=n \cdot M_7-n+M_7+n = M_7.=n⋅M7−n+M7+n=M7.