Question

4. chứng tỏ rằng : 3 mũ 0+ 3 mũ 1+ 3 mũ 2+ 3 mũ 3 ..............+ 3 mũ 11 chia hết cho 40

Answer 1

Ta có: 3^0 + 3^1 + 3^2 + 3^3 + ... + 3^11

= ( 3^0 + 3^1 + 3^2 + 3^3 ) + ... + ( 3^8 + 3^9 + 3^10 + 3^11 )

= 40 + ... + 3^8 . ( 3^0 + 3^1 + 3^2 + 3^3 )

= 40 + ... + 3^8 . 40

= 40 . ( 1 + ... + 3^8 ) \(⋮\)40

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Answer 2

\(1+3+3^2+............+3^{11}\)

\(=\left(1+3+3^2+3^3\right)+\left(3^4+3^5+3^6+3^7\right)+\left(3^8+3^9+3^{10}+3^{11}\right)\)

\(=1\left(1+3+3^2+3^3\right)+3^4\left(1+3+3^2+3^3\right)+3^8\left(1+3+3^2+3^3\right)\)

\(=1.40+3^4.40+3^8.40\)

\(=40\left(1+3^4+3^8\right)⋮40\left(đpcm\right)\)