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

\(\Leftrightarrow A=\left(7+7^2+7^3+7^4\right)+\left(7^5+7^6+7^7+7^8\right)+........+\left(7^{4n-3}+7^{4n-2}+7^{4n-1}+7^{4n}\right)\)

\(\Leftrightarrow A=7\left(1+7+7^2+7^3\right)+7^5\left(1+7+7^2+7^3\right)+........+7^{4n-3}\left(1+7+7^2+7^3\right)\)

\(\Leftrightarrow A=7.400+7^5.400+...........+7^{4n-3}.400\)

\(\Leftrightarrow A=400\left(7+7^5+........+7^{4n-3}\right)⋮400\left(đpcm\right)\)

Ta có:

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

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

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

\(A=7\times400+7^5\times400+...+7^{4n-3}\times400\)

\(A=400\left(7+7^5+...+7^{4n-3}\right)⋮400\)

Ta có 7¹+7²+7³+7⁴+...+7^4n-1+7⁴ⁿ

= (7¹+7²+7³+7⁴)+...+(7^4n-3+7^4n-2+7^4n-1+7⁴ⁿ)

= (7¹+7²+7³+7⁴)+...+7^4n-3(7¹+7²+7³+7⁴)

= 2800+...+7^4n-3.2800

= 2800.(1+...+7^4n-3) 

Mà 2800 chia hết cho 400 nên 7¹+7²+7³+7⁴+...+7^4n-1+7⁴ⁿ chia hết cho 400

\(D=\left(7^1+7^2+7^3+7^4\right)+\left(7^5+7^6+7^7+7^8\right)+...+\left(7^{4n-3}+7^{4n-2}+7^{4n-1}+7^{4n}\right)\)

\(\Rightarrow D=7^1.\left(1+7+7^2+7^3\right)+7^5.\left(1+7+7^2+7^3\right)+...+7^{4n-3}.\left(1+7+7^2+7^3\right)\)

\(\Rightarrow D=7^1.400+7^5.400+...+7^{4n-3}.400=400.\left(7^1+7^5+...+7^{4n-3}\right)\)

Vậy D chia hết cho 400

Ta có:    \(A=7+7^2+7^3+.....+7^{4n}\)                      \(\left(n\in N\right)\)

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

      \(\Leftrightarrow A=7.400+7^5.400+....+7^{4n-3}.400\)

      \(\Leftrightarrow\left(7+7^5+....+7^{4n-3}\right).400\) chia hết cho 400

Vậy A chia hết cho 400

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