Question

Chứng minh rằng : A = \(\left(3+3^2+3^3+..........+3^9\right)\)chia hết cho 13

Answer 1

A=3+3^2+3^3+3^4+...+3^9 
=(3+3^2+3^3) + 3^3(3+3^2+3^3)+3^6(3+3^2+3^3) 
=(3+3^2+3^3).(1+3^3+3^6) 
=3(1+3+3^2)(1+3^3+3^6) 
=3.13.(1+3^3+3^6) chia hết cho 13 

Answer 2

\(A=\left(3+3^2+3^3\right)+...+\left(3^7+3^8+3^9\right)\)

\(=3\left(1+3+3^2\right)+.....+3^7\left(1+3+3^2\right)\)

\(=13\left(3+3^4+3^7\right)⋮13\left(đpcm\right)\)

Answer 3

Thấy A là tổng của 9 số hạng.

\(\implies A=(3+3^2+3^3)+(3^4+3^5+3^6)+(3^7+3^8+3^9) \) (có 3 nhóm)

\(\implies A=3(1+3+3^2)+3^4(1+3+3^2)+3^7(1+3+3^2)\)

\(\implies A=3.13+3^4.13+3^7.13\)

\(\implies A=13(3+3^4+3^7) \vdots13\)

\(\implies A\vdots13\)

Vậy \(A\vdots 13 (đpcm)\)

_Học tốt_