Question

Chứng minh rằng tổng \(A=7^1+7^2+7^3+7^4+....+7^{4k}\); \(k\in N\) chia hết cho 100.

Answer 1

\(A=7^1+7^2+7^3+7^4+...+7^{4k}\)

\(=\left(7^1+7^2+7^3+7^4\right)+...+\left(7^{4k-3}+7^{4k-2}+7^{4k-1}+7^{4k}\right)\)

\(=7.\left(1+7+7^2+7^3\right)+...+7^{4k-3}.\left(1+7+7^2+7^3\right)\)

\(=7.\left(1+7+49+343\right)+...+7^{4k-3}.\left(1+7+49+343\right)\)

\(=7.400+...+7^{4k-3}.400=400.\left(7+...+7^{4k-3}\right)\)

\(=100.\left[4.\left(7+...+7^{4k-3}\right)\right]⋮100\)

=> đpcm