Question

chứng minh rằng :

a. A = 1 + 4 + 4² + 4³ + ......... + 4⁵⁸ + 4^{59 \(⋮\)} 5; 21; 85

b. B = 5 + 5³ + 5⁵ + ......... + 5²⁰² + 5^{203 \(⋮\)} 31

Answer 1

a1. A = \(1+4+4^2+4^3+...+4^{58}+4^{59}\)

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

A = \(5+4^2.5+...+4^{58}.5\)

A = \(5\left(1+4^2+...+4^{58}\right)⋮5\)

a2. A = \(1+4+4^2+4^3+...+4^{58}+4^{59}\)

A = \(\left(1+4+4^2\right)+\left(4^3+4^4+4^5\right)+...+\left(4^{57}+4^{58}+4^{59}\right)\)

A = \(\left(1+4+4^2\right)+4^3\left(1+4+4^2\right)+...+4^{57}\left(1+4+4^2\right)\)

A = \(\left(1+4+4^2\right)\left(1+4^3+...+4^{57}\right)\)

A = \(21.\left(1+4^3+...+4^{57}\right)⋮21\)

a3. A = \(1+4+4^2+4^3+...+4^{58}+4^{59}\)

A = \(\left(1+4+4^2+4^3\right)+\left(4^4+4^5+4^6+4^7\right)+...+\left(4^{56}+4^{57}+4^{58}+4^{59}\right)\)

A = \(\left(1+4+4^2+4^3\right)+4^4\left(1+4+4^2+4^3\right)+...+4^{56}\left(1+4+4^2+4^3\right)\)

A = \(\left(1+4+4^2+4^3\right)\left(1+4^4+...+4^{56}\right)\)

A = \(85.\left(1+4^4+...+4^{56}\right)⋮85\)

Câu B sao thứ tự số mũ chẳng có quy luật vậy, sao mà làm được :v

Answer 2

mình đặt tên cho dễ

A=1 + 4 + 4^2 + ..... + 4 ^59 \(⋮5\)

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

A=5+4^2.5+....4^58.5

A=5.(1+4^2+....+4^58) => đcpm

B=1 + 4 + 4^2 + ..... + 4 ^59 \(⋮21\)

B=(1+4+4^2)+.........+(4^57+4^58+4^59)

B= (1+4+4^2)+4^3(1+4+4^2)+.....+4^47(1+4+4^2

B=(1+4+4^2)+1+4^3+.....+4^57)

B=21.(1+4^3+.....+4^57)\(⋮21\Rightarrowđcpm\)