Question

Cho H = 2+2²+2³+.........+2⁶⁰

Hãy chứng tỏ H chia hêt cho 3, 7 và 15

Answer 1

Ta có:

\(H=2+2^2+2^3+...+2^{60}\)

\(H=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{59}+2^{60}\right)\)

\(H=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{59}\cdot\left(1+2\right)\)

\(H=3\cdot\left(2+2^3+...+2^{59}\right)\)

Vậy H chia hết cho 3

_______

\(H=2+2^2+2^3+...+2^{60}\)

\(H=\left(2+2^2+2^3\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)

\(H=2\cdot\left(1+2+4\right)+2^4\cdot\left(1+2+4\right)+...+2^{58}\cdot\left(1+2+4\right)\)

\(H=7\cdot\left(2+2^4+...+2^{58}\right)\)

Vậy H chia hết cho 7

__________

\(H=2+2^2+2^3+...+2^{60}\)

\(H=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)

\(H=2\cdot\left(1+2+4+8\right)+2^5\cdot\left(1+2+4+8\right)+...+2^{57}\cdot\left(1+2+4+8\right)\)

\(H=15\cdot\left(2+2^5+...+2^{57}\right)\)

Vậy H chia hết cho 15 

Answer 2

 

$H=2+2^2+2^3+\mathrm{...}+2^{60}$

Ta có:

 $H=2.\left(1+2\right)+2^3.\left(1+2\right)+\mathrm{...}+2^{59}.(1+2)$

 $H=2.3+2^3.3+\mathrm{...}+2^{59}.3$

$H=3.\left(2+2^3+\mathrm{...}+2^{59}\right)⋮3$

Ta có:

 $H=2.\left(1+2+2^2\right)+2^4.\left(1+2+2^2\right)+\mathrm{...}+2^{28}.\left(1+2+2^2\right)$

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Answer 3

hoho

 

 

Answer 4

MÌNH KO BIÉT

Answer 5

Bạn nào không biết đừng nhắn bừa!!!