Cho A 2+2^2+2^3+…+2^200 Chứng minh răng A : 7,15,3

Cho A 2+2^2+2^3+…+2^200 Chứng minh răng A : 7,15,3

1 bình luận về “Cho A 2+2^2+2^3+…+2^200 Chứng minh răng A : 7,15,3”

  1. $\text{A = 2 + 2² + 2³ + … + $2^{200}$}$
    $\text{= (2 + 2²) + (2³ + $2^4$) + … + ($2^{199}$ + $2^{200}$)}$
    $\text{= 2 × (1 + 2) + 2³ × (1 + 2) + … + $2^{199}$ × (1 + 2)}$
    $\text{= 2 × 3 + 2³ × 3 + … + $2^{199}$ × 3}$
    $\text{= 3× (2 +  2³ + … + $2^{199}$)}$
    $\text{Vì 3 $\vdots$ 3}$
    $\text{⇒ A $\vdots$ 3}$
    $\text{_____}$
    $\text{A = 2 + 2² + 2³ + … + $2^{200}$}$
    $\text{= (2 + 2² + 2³) + ($2^4$ + $2^5$ + $2^6$) + … + ($2^{198}$ + $2^{199}$ + $2^{200}$)}$
    $\text{2 × (1 + 2 + 2²) + $2^4$ × (1 + 2 + 2²) + … + $2^{198}$ × (1 + 2 + 2²)}$
    $\text{= 2 × 7 + $2^4$ × 7 + … + $2^{198}$ × 7}$
    $\text{= 7 × (2 + $2^4$ + … + $2^{198}$}$
    $\text{Vì 7 $\vdots$ 7}$
    $\text{⇒ A $\vdots$ 7}$
    $\text{_____}$
    $\text{A = 2 + 2² + 2³ + … + $2^{200}$}$
    $\text{= (2 + 2² + 2³ + $2^4$) + ($2^5$ + $2^6$ + $2^7$ + $2^8$) + … + ($2^{197}$ + $2^{198}$ + $2^{199}$ + $2^{200}$)}$
    $\text{= 2 × (1 + 2 + 2² + 2³) + $2^5$ × (1 + 2 + 2² + 2³) + … +$2^{197}$ × (1 + 2 + 2² + 2³)}$
    $\text{= 2 × 15 + $2^5$ × 15 + … +$2^{197}$ × 15}$
    $\text{= 15 × (2 + $2^5$ + … +$2^{197}$)}$
    $\text{Vì 15 $\vdots$ 15}$
    $\text{⇒ A $\vdots$ 15}$

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