Question

1, Chứng minh: A= 2+2^2+2^3+..........................+2^59+2^60

a, A chia hết cho 3

b, A chia hết cho 7

c, A chia hết cho 15

 

 

Answer 1

A = 2 + 2² + 2³ + ... + 2⁶⁰

= (2 + 2²) + (2³ + 2⁴) + ... + (2⁵⁹ + 2⁶⁰)

= 2.(1 + 2) + 2³.(1 + 2) + ... + 2⁵⁹.(1 + 2)

= 2.3 + 2³.3 + ... + 2⁵⁹.3

= 3.(2 + 2³ + ... + 2⁵⁹) chia hết cho 3

A = 2 + 2² + 2³ + ... + 2⁶⁰

= (2 + 2² + 2³) + (2⁴ + 2⁵ + 2⁶) + ... + (2⁵⁸ + 2⁵⁹ + 2⁶⁰)

= 2.(1 + 2 + 2²) + 2⁴.(1 + 2 + 2²) + ... + 2⁵⁸.(1 + 2 + 2²)

= 2.7 + 2⁴.7 + ... + 2⁵⁸.7

= 7.(2 + 2⁴ + ... + 2⁵⁸) chia hết cho 7

A = 2 + 2² + 2³ + ... + 2⁶⁰

= (2 + 2² + 2³ + 2⁴) + (2⁵ + 2⁶ + 2⁷ + 2⁸) + ... + (2⁵⁷ + 2⁵⁸ + 2⁵⁹ + 2⁶⁰)

= 2.(1 + 2 + 2² + 2³) + 2⁵.(1 + 2 + 2² + 2³) + ... + 2⁵⁷.(1 + 2 + 2² + 2³)

= 2.15 + 2⁵.15 + ... + 2⁵⁷.15

= 15.(2 + 2⁵ + ... + 2⁵⁷) chia hết cho 15

Answer 2

bạn có nhầm đề ko ? Xem lại 2⁵⁰ +2⁶⁰ 

 

Answer 3

a)A= 2+2^2+2^3+...+2^59+2^60

A=(2+2^2)+(2^3+2^4)+(2^5+2^6)+...+(2^59+2^60)

A=2.(1+2)+2^3.(1+2)+2^5.(1+2)+...+2^59.(1+2)

A=2.3+2^3.3+2^5.3+...+2^59.3

A=3.(2+2^3+2^5+...+2^59)

=>A chia hết cho 3

Vậy A chia hết cho 3

b)A= 2+2^2+2^3+..........................+2^59+2^60

A=(2+2^2+2^3)+(2^4+2^5+2^6)+(2^7+2^8+2^9)+...+(2^58+2^59+2^60)

A=2.(1+2+2^2)+2^4.(1+2+2^2)+2^7.(1+2+2^2)+...+2^58.(1+2+2^2)

A=2.7+2^4.7+2^7.7+...+2^58.7

A=7.(2+2^4+2^7+...+2^58)

=>A chia hết cho 7

Vậy A chia hết cho 7

c)A= 2+2^2+2^3+..........................+2^59+2^60

A=(2+2^2+2^3+2^4)+(2^5+2^6+2^7+2^8)+(2^9+2^10+2^11+2^12)+...+(2^57+2^58+2^59+2^60)

A=2.(1+2+2^2+2^3)+2^5.(1+2+2^2+2^3)+2^9.(1+2+2^2+2^3)+...+2^57.(1+2+2^2+2^3)

A=2.15+2^5.15+2^9.15+...+2657.15

A=15.(2+2^5+2^9+...+2^57)

=>A chia hết cho 15

Vậy A chia hết cho15