Question

Cho M= 2+2²+2³+..+2²⁰

Chứng tỏ rằng M chia hết cho 5

Answer 1

Ta có: M = 2+2²+2³+....+2²⁰

    => M = (2+2²+2³)+(2⁴+2⁵+2⁶)+...+(2¹⁷+2¹⁸+2¹⁹+2²⁰⁾

      => M = 2 x (1+2+2²) + 2⁴ x (1+2+2²)+....+2¹⁷ x (1+2+2²)

     => M = 2 x 5 + 2⁴ x 5 +......+2¹⁷ x 5

     => M = 5 x (2+2⁴+...+2¹⁷) chia hết cho 5

Vậy M chia hết cho 5

Answer 2

M=2+2²+2³+...+2^20.

  =(2+2²+2³+2⁴)+(2⁵+2⁶+2⁷+2⁸)+...+(2¹⁷+2¹⁸+2¹⁹+2²⁰).

  =2.(1+2+2²+2³)+2⁵.(1+2+2²+2³)+...+2¹⁷.(1+2+2²+2³).

  =2.15+2⁵+15+...+2¹⁷+15.

   =15.2.(1+2⁴+...+2¹⁶)

=3.5.2.(1+2⁴+...+2¹⁶) ^{chia hết cho 5}

  

Answer 3

\(M=2+2^2+2^3+...............+2^{20}\)

\(\Leftrightarrow M=\left(2+2^2+2^3+2^4\right)+....................+\left(2^{17}+2^{18}+2^{19}+2^{20}\right)\)(có 5 cặp)

\(\Leftrightarrow M=2.\left(1+2+2^2+2^3\right)+....................+2^{17}.\left(1+2+2^2+2^3\right)\)

\(\Leftrightarrow M=\left(2+..............+2^{17}\right).\left(1+2+2^2+2^3\right)\)

\(\Leftrightarrow M=\left(2+....................+2^{17}\right).15⋮5\)(ĐPCM)

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