Question

 Cho A=3^101 + 3^102 +3^103 + ..... + 3^200

CMR :  A chia hết cho 120

 

Answer 1

\(A=3^{101}+3^{102}+3^{103}+...+3^{200}\)

\(3A=3^{102}+3^{103}+3^{104}+...+3^{201}\)

\(3A-A=\left(3^{102}+3^{103}+3^{104}+3^{201}\right)-\left(3^{101}+3^{102}+3^{103}+...+3^{201}\right)\)

\(2A=3^{201}-3^{101}\)

\(2A=3^{100}\)

\(\Rightarrow A=3^{100}:2\)

Answer 2

\(A=3^{101}+3^{102}+3^{103}+...+3^{200}\)

\(A=3^{101}+3^{102}+3^{103}+3^{104}+...+3^{197}+3^{198}+3^{199}+3^{200}\)

\(A=3^{100}\left(3+3^2+3^3+3^4\right)+...+3^{196}\left(3+3^2+3^3+3^4\right)\)

\(A=120\left(3^{100}+...+3^{196}\right)⋮120\)