Question

chứng minh A = 2^1+ 2^2+2^3+2^4+2^5+.....+2^2010 chia hết cho 3 và 7

Answer 1

A=2+2^2+2^3+2^4+2^5+....+2^2010

=(2+2^2)+(2^3+2^4)+(2^5+2^6)+....+(2^2009+2^2010)

=2(1+2)+2^3(1+2)+2^5(1+2)+....+2^2009(1+2)

=(1+2)(2+2^3+2^5+....+2^2009)

=3(2+2^3+2^5+...+2^2009) chia hết cho 3

A=2+2^2+2^3+2^4+2^5+...+2^2010

=(2+2^2+2^3)+(2^4+2^5+2^6)+...+(2^2008+2^2009+2^2010)

=2(1+2+2^2)+2^4(1+2+2^2)+...+2^2008(1+2+2^2)

=7(2+2^4+...+2^2008) chia hết cho 7