Question

[dao thi yen nhi]

CHỨNG MINH TỔNG SAU CHIA HẾT CHO 65 

B=5+5^2+5^3+......+5^2012

Answer 1

[le bao truc]

\(B=\left(5+5^2+5^3+5^4\right)+\left(5^5+5^6+5^7+5^8\right)+\left(5^{2009}+5^{2010}+5^{2011}+5^{2012}\right)\)
\(B=\left(5+5^2+5^3+5^4\right)+5^4\left(5+5^2+5^3+5^4\right)+...+5^{2008}\left(5+5^2+5^3+5^4\right)\)
\(B=\left(5+5^2+5^3+5^4\right).\left(1+5^4+...+5^{2008}\right)\)
\(B=780\left(1+5^4+...+5^{2008}\right)⋮65\left(780⋮65\right)\)

Answer 2

[Nguyễn Thành Thái]

B=(5+5^2+5^3+5^4)+(5^5+5^6+5^7+5^8)+...+(5^2009+5^2010+5^2011+5^2012)

B=(5+5^2+5^3+5^4)+5^4(5+5^2+5^3+5^4)+...+5^2008(5+5^2+5^3+5^4)

B=(5+5^2+5^3+5^4).(1+5^4+5^5+...+5^2008)

B=780(1+5^4+5^5+...+5^2008)

Vì 780 chia hết cho 65

suy ra 5+5^2+5^3+5^4+5^5+...+5^2012) chia hết cho 65