Question

chứng minh B=7+7^2+7^3+...+7^100 chia hết cho 8 và 399

Answer 1

\(B=7+7^2+...+7^{100}\)

\(B=\left(7+7^2\right)+...+\left(7^{99}+7^{100}\right)\)

\(B=7\left(1+7\right)+...+7^{99}\left(1+7\right)\)

\(B=7\cdot8+...+7^{99}\cdot8\)

\(B=8\cdot\left(7+...+7^{99}\right)⋮8\left(đpcm\right)\)

Answer 2

\(B=7+7^2+7^3+...+7^{100}\)

\(B=\left(7+7^2+7^3\right)+...+\left(7^{98}+7^{99}+7^{100}\right)\)

\(B=399\cdot1+...+7^{97}\cdot\left(7+7^2+7^3\right)\)

\(B=399\cdot1+...+7^{97}\cdot399\)

\(B=399\cdot\left(1+...+7^{97}\right)⋮399\left(đpcm\right)\)

Answer 3

\(B=7+7^2+7^3+7^4+7^5+7^6+....+7^{100}\)

     \(=\left(7+7^2\right)+\left(7^3+7^4\right)+...+\left(7^{99}+7^{100}\right)\)

     \(=7.\left(1+7\right)+7^3.\left(1+7\right)+...+7^{99}.\left(1+7\right)\)

       \(=7.8+7^3.8+...+7^{99}.8\)

       \(=8.\left(7+7^3+...+7^{99}\right)⋮8\)

Vậy B chia hết cho 8

Ta lại có: \(B=\left(7+7^2+7^3\right)+\left(7^4+7^5+7^6\right)+...+\left(7^{98}+7^{99}+7^{100}\right)\)

                   \(=\left(7+7^2+7^3\right)+7^3.\left(7+7^2+7^3\right)+....+7^{97}.\left(7+7^2+7^3\right)\)

                     \(=399+7^3.399+.....+7^{97}.399\)

                      \(=399.\left(1+7^3+....+7^{97}\right)⋮399\)

Vậy B chia hết cho 399

         

Answer 4

B = (7+7^2) + (7^3+7^4) +...+ (7^99+7^100)

B = 7.(7+1) + 7^3.(7+1) +...+ 7^99.(7+1)

B = 7.8 + 7^3.8 +...+ 7^99.8

B = 8.(7+7^3+...+7^99) chia hết cho 8

Vậy B chia hết cho 8

Answer 5

B = (7+7^2+7^3) +...+ (7^98+7^99+7^100)

B = 7.(7+7^2+7^3) +...+ 7^98.(7+7^2+7^3)

B = 7.399 +...+ 7^98.399

B = 399.(7+...+7^98) chia hết cho 399

Vậy B chia hết cho 399

Answer 6

*) \(B=7+7^2+7^3+...+7^{100}\)

\(\Rightarrow B=\left(7+7^2\right)+\left(7^3+7^4\right)+...+\left(7^{99}+7^{100}\right)\)

\(\Rightarrow B=7\left(1+7\right)+7^3\left(1+7\right)+...+7^{99}\left(1+7\right)\)

\(\Rightarrow B=7\cdot8+7^3\cdot8+...+7^{99}\cdot8\)

\(\Rightarrow B=8\left(7+7^3+...+7^{99}\right)\)

\(\Rightarrow B⋮8\left(đpcm\right)\)

*) \(B=7+7^2+7^3+...+7^{100}\)

\(\Rightarrow B=\left(7+7^2+7^3\right)+\left(7^4+7^5+7^6\right)+...+\left(7^{98}+7^{99}+7^{100}\right)\)

\(\Rightarrow B=399+7^3\left(7+7^2+7^3\right)+...+7^{97}\left(7+7^2+7^3\right)\)

\(\Rightarrow B=399+7^3\cdot399+...+7^{97}\cdot399\)

\(\Rightarrow B=399\left(1+7^3+...+7^{97}\right)\)

\(\Rightarrow B⋮399\left(đpcm\right)\)