Question

cho A=2²+2³+2⁴+...+2²⁰

^{Chứng tỏ rằng A+4 không là số chính phương}

Answer 1

\(A=4+\left(2^2+2^3+2^4+...+2^{20}\right)\)  

\(A-4=2^2+2^3+2^4+...+2^{20}\) 

\(2\left(A-4\right)=2^3+2^4+2^5+...+2^{21}\)

\(A-4=2\left(A-4\right)-\left(A-4\right)=\left(2^3+2^4+2^5+...+2^{21}\right)-\left(2^2+2^3+2^4+...+2^{20}\right)\) 

\(A-4=\left(2^3-2^3\right)+\left(2^4-2^4\right)+...+\left(2^{20}-2^{20}\right)+\left(2^{21}-2^2\right)\)

\(A-4=\left(2^{21}-4\right)\)

\(A=\left(2^{21}-4+4\right)\)

\(A=2^{21}\)