Toán CMR 2 +1 + $2^{2}$ + $2^{3 }$ + … + $2^{7}$ chia hết cho 3 25/11/2021 By Anna CMR 2 +1 + $2^{2}$ + $2^{3 }$ + … + $2^{7}$ chia hết cho 3
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 Trả lời
`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` Trả lời
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
`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`