Question

Chứng minh rằng tổng sau chia hết cho 5 :

\(s=2+2^2+2^3+2^4+...+2^{2012}\)

Answer 1

Ta có: 

= (2+2²)+(2³+2⁴)+....+(2²⁰¹¹+2²⁰¹²)

=2(1+2²)+2³(1+2²)+....+2²⁰¹¹(1+2²)

=2.5+2³.5+....+2²⁰¹¹.5

=5.(2+2³+....+2²⁰¹¹)

Vậy S chia hất cho 5

 

 

Answer 2

\(S=\left(2+2^3\right)+\left(2^2+2^4\right)+..........+\left(2^{2010}+2^{2012}\right)\)

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

\(=5.\left(2+2^2+.........+2^{2010}\right)\)

=> S chia hết cho 5