Ta có:
\(1+7+7^2+7^3+...+7^{100}+7^{101}\)
\(=\left(1+7\right)+\left(7^2+7^3\right)+...+\left(7^{100}+7^{101}\right)\)
\(=1.\left(1+7\right)+7^2.\left(1+7\right)+...+7^{100}.\left(1+7\right)\)
\(=1.8+7^2.8+...+7^{100}.8\)
\(=8.\left(1+7^2+...+7^{100}\right)\)
\(\Rightarrow1+7+7^2+7^3+...+7^{100}+7^{101}⋮8\)
\(=\left(1+7\right)+\left(7^2+7^3\right)+...+\)\(\left(7^{100}+7^{101}\right)\)
\(=8+7^2.\left(1+7\right)+...+\)\(7^{100}.\left(1+7\right)\)
\(=8+7^2.8+...+7^{100}.8\)
\(=8.\left(1+7^2+...+7^{100}\right)\)chia hết cho 8 (đpcm)
1+7+72+73+...+7100+7101=(1+7)+(72+73)+...+(7100+7101)
=(1+7)+72(1+7)+...+7100(1+7)
= 8 + 72.8+...+7100.8
=8(1+72+...+7100)
Vì 8 chia hết cho 8 =>8(1+72+...+7100) chia vậy cho 8=>1+7+72+73+...+7100+7101 chia hết cho 8
Vậy 1+7+72+73+...+7100+7101chia hết cho 8
Đặt A= 1 + 7 + 72 + 73 +...+ 7100 + 7101
->A= (1 + 7) +72 *(1 + 7) + 74 * (1+ 7)+....+ 7100 *(1+7)
->A= 8 + 72 *8 +74 *8+....+7100 *8
->A=8*(1+72 +74+....7100) chia hết cho 8
-> A chia hết cho 8(điều phải chứng minh) (* là dấu nhân bạn nhé)
