a) \(A=2+2^2+2^3+\dots+2^{60}\)

\(2A=2^2+2^3+2^4+\dots+2^{61}\)

\(2A-A=\left(2^2+2^3+2^4+\dots+2^{61}\right)-\left(2+2^2+2^3+\dots+2^{60}\right)\)

\(A=2^{61}-2\)

Vậy: \(A=2^{61}-2\).

b)

+) \(A=2+2^2+2^3+\dots+2^{60}\)

\(=\left(2+2^2\right)+\left(2^3+2^4\right)+\left(2^5+2^6\right)+\dots+\left(2^{59}+2^{60}\right)\)

\(=2\cdot\left(1+2\right)+2^3\cdot\left(1+2\right)+2^5\cdot\left(1+2\right)+\dots+2^{59}\cdot\left(1+2\right)\)

\(=2\cdot3+2^3\cdot3+2^5\cdot3+\dots+2^{59}\cdot3\)

\(=3\cdot\left(2+2^3+2^5+\dots+2^{59}\right)\)

Vì \(3\cdot\left(2+2^3+2^5+\dots+2^{59}\right)⋮3\) nên \(A⋮3\)

+) \(A=2+2^2+2^3+\dots+2^{60}\)

\(=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+\left(2^9+2^{10}+2^{11}+2^{12}\right)+\dots+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)

\(=2\cdot\left(1+2+2^2+2^3\right)+2^5\cdot\left(1+2+2^2+2^3\right)+2^9\cdot\left(1+2+2^2+2^3\right)+\dots+2^{57}\cdot\left(1+2+2^2+2^3\right)\)

\(=2\cdot15+2^5\cdot15+2^9\cdot15+\dots+2^{57}\cdot15\)

\(=15\cdot\left(2+2^5+2^9+\dots+2^{57}\right)\)

Vì \(15⋮5\) nên \(15\cdot\left(2+2^5+2^9+\dots+2^{57}\right)⋮5\)

hay \(A\vdots5\)

+) \(A=2+2^2+2^3+\dots+2^{60}\)

\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+\left(2^7+2^8+2^9\right)+\dots+\left(2^{58}+2^{59}+2^{60}\right)\)

\(=2\cdot\left(1+2+2^2\right)+2^4\cdot\left(1+2+2^2\right)+2^7\cdot\left(1+2+2^2\right)+\dots+2^{58}\cdot\left(1+2+2^2\right)\)

\(=2\cdot7+2^4\cdot7+2^7\cdot7+\dots+2^{58}\cdot7\)

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

Vì \(7\cdot\left(2+2^4+2^7+\dots+2^{58}\right)⋮7\) nên \(A⋮7\)

$Toru$

a) $A=2+2^2+2^3+\cdots +2^{60}$

$2A=2^2+2^3+2^4+\cdots +2^{61}$

$2A-A=\left(2^2+2^3+2^4+\cdots +2^{61}\right)-\left(2+2^2+2^3+\cdots +2^{60}\right)$

$A=2^{61}-2$

Vậy: $A=2^{61}-2$.

b)

+) $A=2+2^2+2^3+\cdots +2^{60}$

$=\left(2+2^2\right)+\left(2^3+2^4\right)+\left(2^5+2^6\right)+\cdots +\left(2^{59}+2^{60}\right)$

$=2\cdot \left(1+2\right)+2^3\cdot \left(1+2\right)+2^5\cdot \left(1+2\right)+\cdots +2^{59}\cdot \left(1+2\right)$

$=2\cdot 3+2^3\cdot 3+2^5\cdot 3+\cdots +2^{59}\cdot 3$

$=3\cdot \left(2+2^3+2^5+\cdots +2^{59}\right)$

Vì $3\cdot \left(2+2^3+2^5+\cdots +2^{59}\right)⋮3$ nên $A⋮3$

+) $A=2+2^2+2^3+\cdots +2^{60}$

$=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+\left(2^9+2^{10}+2^{11}+2^{12}\right)+\cdots +\left(2^{57}+2^{58}+2^{59}+2^{60}\right)$

$=2\cdot \left(1+2+2^2+2^3\right)+2^5\cdot \left(1+2+2^2+2^3\right)+2^9\cdot \left(1+2+2^2+2^3\right)+\cdots +2^{57}\cdot \left(1+2+2^2+2^3\right)$

$=2\cdot 15+2^5\cdot 15+2^9\cdot 15+\cdots +2^{57}\cdot 15$

$=15\cdot \left(2+2^5+2^9+\cdots +2^{57}\right)$

Vì $15⋮5$ nên $15\cdot \left(2+2^5+2^9+\cdots +2^{57}\right)⋮5$

hay $A⋮5$

+) $A=2+2^2+2^3+\cdots +2^{60}$

$=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+\left(2^7+2^8+2^9\right)+\cdots +\left(2^{58}+2^{59}+2^{60}\right)$

$=2\cdot \left(1+2+2^2\right)+2^4\cdot \left(1+2+2^2\right)+2^7\cdot \left(1+2+2^2\right)+\cdots +2^{58}\cdot \left(1+2+2^2\right)$

$=2\cdot 7+2^4\cdot 7+2^7\cdot 7+\cdots +2^{58}\cdot 7$

$=7\cdot \left(2+2^4+2^7+\cdots +2^{58}\right)$

Vì $7\cdot \left(2+2^4+2^7+\cdots +2^{58}\right)⋮7$ nên $A⋮7$

Mình chỉ làm được ý 3 thôi: 

A = 2¹ + 2² + 2³ + ................ + 2¹²⁰

Chứng minh chia hết cho 7

A = 2¹ + 2² + 2³ + ................ + 2¹²⁰

A = (2¹ + 2² + 2³) + (2⁴ + 2⁵ + 2⁶) + ................ + (2¹¹⁸ + 2¹¹⁹ + 2¹²⁰)

A = 2.(1 + 2 + 4) + 2⁴.(1 + 2 + 4) + ................. + 2¹¹⁸.(1 + 2 + 4)

A = 2.7 + 2⁴ . 7 + ................ + 2¹¹⁸.7

A = 7.(2 + 2⁴ + ........... + 2¹¹⁸)

Chứng minh chia hết cho 31

A = 2¹ + 2² + 2³ + ................ + 2¹²⁰ 

A = (2¹ + 2² + 2³ + 2⁴ + 2⁵) + (2⁶ + 2⁷ + 2⁸ + 2⁹ + 2¹⁰) + ................ + (2¹¹⁶ + 2¹¹⁷ + 2¹¹⁸ + 2¹¹⁹ + 2¹²⁰)

A = 2.(1 + 2 + 4 + 8 + 16) + 2⁶.(1 + 2 +4 + 8 + 16) + ............. + 2¹¹⁶.(1 + 2 + 4 + 8 + 16)

A = 2.31 + 2⁶.31 + ....... + 2¹¹⁶ . 31

A = 31.(2 + 2⁶ + ........... + 2¹¹⁶)

 A= (2¹+2²+2³)+(2⁴+2⁵+2⁶)+...+(2⁵⁸+2⁵⁹+2⁶⁰)

   =2⁰(2¹+2²+2³)+2³(2¹+2²+2³)+...+2⁵⁷(2¹+2²+2³)

   =(2¹+2²+2³)(2⁰+2³+...+2⁵⁷⁾

   =     14(2⁰+2³+...+2⁵⁷) chia hết cho 7

Vậy A chia hết cho 7     

gọi 1/41+1/42+1/43+...+1/80=S

ta có :

S>1/60+1/60+1/60+...+1/60

S>1/60 x 40

S>8/12>7/12

Vậy S>7/12

a) A = 2 + 2² + 2³ + ... + 2¹² (có 12 số; 10 chia hết cho 2)

A = (2 + 2²) + (2³ + 2⁴) + ... + (2¹¹ + 2¹²)

A = 2.(1 + 2) + 2³.(1 + 2) + ... + 2¹¹.(1 + 2)

A = 2.3 + 2³.3 + ... + 2¹¹.3

A = 3.(2 + 2³ + ... + 2¹¹)

Vì 2 + 2³ + ... + 2¹¹ chia hết cho 2 => A chia hết cho 3.2 = 6 (đpcm)

b) bn xem lại đề -_-, A ko chia hết cho 12

Vì 2 + 2³ + ... + 2¹¹ chia hết cho 2 nhưng không chia hết cho 4

b) Dựa vào câu a ta đã có được

A = 3.(2 + 2³ + ... + 2¹¹)

Do các lũy thừa của 2 từ 2² trở đi đều chia hết cho 4

=> 2³; 2⁵; ...; 2¹¹ chia hết cho 4

Mà 2 không chia hết cho 4

=> 2 + 2³ + ... + 2¹¹ không chia hết cho 4

=> A không chia hết cho 3.4 = 12

1. \(A=2^{2016}-1\)

\(2\equiv-1\left(mod3\right)\\ \Rightarrow2^{2016}\equiv1\left(mod3\right)\\ \Rightarrow2^{2016}-1\equiv0\left(mod3\right)\\ \Rightarrow A⋮3\)

\(2^{2016}=\left(2^4\right)^{504}=16^{504}\)

16 chia 5 dư 1 nên 16^504 chia 5 dư 1

=> 16^504-1 chia hết cho 5

hay A chia hết cho 5

\(2^{2016}-1=\left(2^3\right)^{672}-1=8^{672}-1⋮7\)

lý luận TT trg hợp A chia hết cho 5

(3;5;7)=1 = > A chia hết cho 105

2;3;4 TT ạ !!

A = 2 + 2² + 2³ + ... + 2²⁰

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

= 30 + 2⁴.(2 + 2² + 2³ + 2⁴) + ... + 2¹⁶.(2 + 2² + 2³ + 2⁴)

= 30 + 2⁴.30 + ... + 2¹⁶.30

= 30.(1 + 2⁴ + ... + 2¹⁶)

= 5.6.(1 + 2⁴ + ... + 2¹⁶) ⋮ 5

Vậy A ⋮ 5

b) A = 2 + 2² + 2³ + ... + 2¹⁰⁰

= (2 + 2² + 2³ + 2⁴) + (2⁵ + 2⁶ + 2⁷ + 2⁸) + ... + (2⁹⁷ + 2⁹⁸ + 2⁹⁹ + 2¹⁰⁰)

= 30 + 2⁴.(2 + 2² + 2³ + 2⁴) + ... + 2⁹⁶.(2 + 2² + 2³ + 2⁴)

= 30 + 2⁴.30 + ... + 2⁹⁶.30

= 30.(1 + 2⁴ + ... + 2⁹⁶)

= 6.5.(1 + 2⁴ + ... + 2⁹⁶) ⋮ 6

Vậy A ⋮ 6

a) \(A=2+2^2+...+2^{120}\)

\(\Rightarrow A=\left(2+2^2\right)+...+\left(2^{119}+2^{120}\right)\)

\(\Rightarrow A=\left(2+2^2\right)+...+2^{118}.\left(2+2^2\right)\)

\(\Rightarrow A=6+...+2^{118}.6\)

\(\Rightarrow A=6.\left(1+...+2^{118}\right)⋮3\Rightarrow A⋮3\left(đpcm\right)\)

b) \(A=2+2^2+...+2^{120}\)

\(\Rightarrow A=\left(2+2^2+2^3\right)+...+\left(2^{118}+2^{119}+2^{120}\right)\)

\(\Rightarrow A=\left(2+2^2+2^3\right)+...+2^{117}.\left(2+2^2+2^3\right)\)

\(\Rightarrow A=14+...+2^{117}.14\)

\(\Rightarrow A=14.\left(1+...+2^{117}\right)⋮7\Rightarrow A⋮7\left(đpcm\right)\)

Chỉnh đề:

Ta có:

\(A=2+2^2+2^3+2^4+...2^{12}\)

\(A=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{10}+2^{11}+2^{12}\right)\)

\(A=14+2^3.\left(2+2^2+2^3\right)+...+2^9.\left(2+2^2+2^3\right)\)

\(A=14+2^3.14+...+2^9.14\)

\(A=14.\left(1+2^3+...+2^9\right)\)

Vì \(14⋮7\) nên \(14.\left(1+2^3+...2^9\right)⋮7\)

Vậy \(A⋮7\)

giỏi dữ ta