Question

Cho P = 4+4²+4³+4⁴+...+4¹¹+4¹² chứng minh rằng P chia hết cho 21

 

Answer 1

Có : P = (4+4^2+4^3)+(4^4+4^5+4^6)+....+(4^10+4^11+4^12)

= 4.(1+4+4^2)+4^4.(1+4+4^2)+....+4^10.(1+4+4^2)

= 4.21+4^4.21+....+4^10.21

= 21.(4+4^4+...+4^10) chia hết cho 21

=> ĐPCM 

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Answer 2

\(P=4+4^2+4^3+4^4+...+4^{11}+4^{12}\)

\(P=4\cdot1+4.4+4.16+4^4\cdot1+...+4^{10}\cdot1+4^{10}\cdot4+4^{10}\cdot16\)

\(P=4\left(1+4+16\right)+4^4\left(1+4+16\right)+...+4^{10}\left(1+4+16\right)\)

\(P=4\cdot21+4^4\cdot21+4^7\cdot21+4^{10}\cdot21\)

\(P=21\left(4+4^4+4^7+4^{10}\right)\)

Vi  \(21⋮21\Rightarrow P⋮21\)