Question

Tổng 3^1+3^2+3^3+3^4+3^5+....+3^2012 có chia hết cho 120 không ? Vì sao?

 

 

Answer 1

\(3^1+3^2+3^3+3^4+3^5+...+3^{2012}\)

\(=\left(3^1+3^2+3^3+3^4\right)+\left(3^5+3^6+3^7+3^8\right)+...+\left(3^{2009}+3^{2010}+3^{2011}+3^{2012}\right)\)

\(=\left(3^1+3^2+3^3+3^4\right)+3^4\left(3^1+3^2+3^3+3^4\right)+...+3^{2008}\left(3^1+3^2+3^3+3^4\right)\)

\(=120\left(1+3^4+...+3^{2008}\right)⋮120\)