CMR 2 +1 + $2^{2}$ + $2^{3 }$ + … + $2^{7}$ chia hết cho 3

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1. Chứng minh rằng :2 +1 + 2² + 2³ + … + $2^{7}$  chia hết cho 3

   2 +1 + 2² + 2³ + … + $2^{7}$

   = 1 + 2+ 2² + 2³ + … + $2^{7}$

   = (1+2) + 2².(1+2) + 2³ . (1+2)+…+ $2^{6}$  . (1+2)

   =3+ 2² . 3 + 2³ . 3 +…. + $2^{6}$ . 3

   Mà 3 ⋮3 ;2² ⋮ 3 ; 2³ ⋮ 3;…

   => 2 +1 + 2² + 2³ + … + $2^{7}$ ⋮ 3

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2. `1“+“2“+“2²“+“2³“+`$2^{4}$+$2^{5}$+$2^{6}$+$2^{7}$

   `=` `(“1“+“2“)“+“(“2²“+“2³“)“+“(`$2^{4}$`+`$2^{5}$)`+“(`$2^{6}$`+`$2^{7}$`)`

   `=“3“+`$2^{2}$`.“(“1“+“2“)“+`$2^{4}$`.“(“1“+“2)“+`$2^{6}$`.“(“1“+“2“)`

   `=“3“+`$2^{2}$`.“3“+`$2^{4}$`.“3“+`$2^{6}$`.“3`

   `=` `3“.“(“1“+`$2^{2}$`+`$2^{4}$`+`$2^{6}$`)`  ⋮  `3`

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