Question

Cho \(A=3^{2013}+3^{2012}+...+3^{2000}\)

Chứng minh rằng A chia hết cho 12

Answer 1

Ta có:

\(A=3^{2000}+...+3^{2012}+3^{2013}⋮3\left(1\right)\)

Lại có:

\(A=3^{2000}+3^{2001}...+3^{2012}+3^{2013}\)

\(\Rightarrow A=\left(3^{2000}+3^{2001}\right)+...+\left(3^{2012}+3^{2013}\right)\)

\(\Rightarrow A=3^{2000}\left(1+3\right)+...+3^{2012}\left(1+3\right)\)

\(\Rightarrow A=3^{2000}.4+...+3^{2012}.4⋮4\left(2\right)\)

Từ (1) và (2)

\(\Rightarrow A=3^{2000}+...+3^{2012}+3^{2013}⋮12\left(đpcm\right)\)