Question

Chứng minh:

A = 2¹ + 2² + 2³ +..............+2²⁰¹⁰ chia hết cho 3; và 7

 

Answer 1

\(A=2^1+2^2+...+2^{2010}\)

\(=\left(2^1+2^2\right)+...+\left(2^{2009}+2^{2010}\right)\)

\(=2^1\left(1+2\right)+...+2^{2009}\left(1+2\right)\)

\(=2^1\cdot3+...+2^{2009}\cdot3\)

\(=3\left(2^1+...+2^{2009}\right)⋮3\)

\(A=2^1+2^2+...+2^{2010}\)

\(=\left(2^1+2^2+2^3\right)+...+\left(2^{2008}+2^{2009}+2^{2010}\right)\)

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

\(=2^1\cdot7+...+2^{2008}\cdot7\)

\(=7\left(2^1+...+2^{2008}\right)⋮7\)