Question

Chứng minh rằng:S=3+3²⁺3³+3⁴⁺....+3⁸⁹+3⁹⁰ chia hết cho 13

Answer 1

\(S=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{88}+3^{89}+3^{90}\right)\)

\(=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+...+3^{88}\left(1+3+3^2\right)\)

\(=3.13+3^4.13+...+3^{88}.13\)

\(=13\left(3+3^4+...+3^{88}\right)\) chia hết cho \(13\)

Answer 2

\(S=3+3^2+3^3+3^4+....+3^{89}+3^{90}\)

\(=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{88}+3^{89}+3^{90}\right)\)

\(==3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+3^{88}\left(1+3+3^2\right)\)

\(=\left(1+3+3^2\right).\left(3+3^4+....+3^{88}\right)\)

\(=13\left(3+3^4+...+3^{88}\right)\)\(⋮\)\(13\)

Answer 3

\(S=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{88}+3^{89}+3^{90}\right)\)

\(=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{88}+3^{89}+3^{90}\right)\)

\(=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+3^{88}\left(1+3+3^2\right)\)

\(=\left(1+3+3^2\right).\left(3+3^4+...+3^{88}\right)\)

\(=13\left(3+3^4+...+3^{88}\right)⋮13\left(đpcm\right)\)