Question

[Hoàng Thúy An]

chung minh rang A=\(7+7^2+7^3+...+7^{2016}⋮20^2\)

giup minh nhe minh dang can gap

Answer 1

[Nguyễn Nam]

\(A=7+7^2+7^3+........+7^{2016}\)

\(A=7\left(1+7+7^2+7^3+........+7^{2012}+7^{2013}+7^{2014}+7^{2015}\right)\)

\(A=7\left[\left(1+7+7^2+7^3\right)+........+\left(7^{2012}+7^{2013}+7^{2014}+7^{2015}\right)\right]\)

\(A=7\left[\left(1+7+7^2+7^3\right)+........+7^{2012}\left(1+7+7^2+7^3\right)\right]\)

\(A=7\left[400+........+7^{2012}.400\right]\)

\(A=7.400\left(1+7^4+7^8+7^{12}+......+7^{2012}\right)⋮400\)

Vì \(20^2=400\) nên \(A⋮20^2\left(dpcm\right)\)