Ta có : 2 + 22 + 23 + ..... + 230
= (2 + 22 + 23) + ..... + (228 + 229 + 230)
= 2.(1 + 2 + 22) + ...... + 228(1 + 2 + 22)
= 2.7 + ..... + 228.7
= 7(2 + ..... + 228) chia hết cho 7
2+22+23+24+...+230=(2+22+23)+(24+25+26)+...+(228+229+230)
= 2(1+2+22)+24(1+2+22)+...+228(1+2+22)=
= (1+2+22)(2+24+...+228)=7.(2+24+...+228) => Chia hết cho 7
\(2+2^2+2^3+2^4+...+2^{28}+2^{29}+2^{30}\)
\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{28}+2^{29}+2^{30}\right)\)
\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+...+2^{28}\left(1+2+2^2\right)\)
\(=2.7+2^4.7+...+2^{28}.7\)
\(=7\left(2+2^4+...+2^{28}\right)\) chia hết cho 7
