Câu 3:

\(A=3+3^2+...+3^{100}\)

\(3A=3^2+3^3+...+3^{101}\)

\(3A-A=3^2+3^3+...+3^{101}-\left(3+3^2+...+3^{100}\right)\)

\(2A=3^{101}-3\) 

Mà: \(2A+3=3^N\)

\(\Rightarrow3^{101}-3+3=3^N\)

\(\Rightarrow3^{101}=3^N\)

\(\Rightarrow N=101\)

Vậy: ... 

Câu 1:

\(A=4+2^2+...+2^{20}\)

Đặt \(B=2^2+2^3+...+2^{20}\)

=>\(2B=2^3+2^4+...+2^{21}\)

=>\(2B-B=2^3+2^4+...+2^{21}-2^2-2^3-...-2^{20}\)

=>\(B=2^{21}-4\)

=>\(A=B+4=2^{21}-4+4=2^{21}\) là lũy thừa của 2

Câu 6:

Đặt A=1+2+3+...+n

Số số hạng là \(\dfrac{n-1}{1}+1=n-1+1=n\left(số\right)\)

=>\(A=\dfrac{n\left(n+1\right)}{2}\)

=>\(A⋮n+1\)

Câu 5:

\(A=5+5^2+...+5^8\)

\(=\left(5+5^2\right)+\left(5^3+5^4\right)+\left(5^5+5^6\right)+\left(5^7+5^8\right)\)

\(=\left(5+5^2\right)+5^2\left(5+5^2\right)+5^4\left(5+5^2\right)+5^6\left(5+5^2\right)\)

\(=30\left(1+5^2+5^4+5^6\right)⋮30\)

a) A chia hết cho 2 vì tất cả các số hạng của tổng đều chia hết cho 2.

b) Ta tách ghép các số hạng của A thành các nhóm sao cho mỗi nhóm xuất hiện thừa số chia hết cho 3. Khi đó:

4₁ A= 2 +2²³ +2 ² + + 220 a₁ A = 2₁ [1 + 2 +2²¹ +. +2¹2):2 Vay A chia hết choi b₁ A = 2 + 2² +2²+ + 220 (2 +2²) + (2 ² + 2 9) + . + (219+220) = 2₁ (1 + 2) + 2² (2+1). .. +2 19 (2+1) + = 2₁3 + 2³.3 + ..+ 219.3. = (2+2 ³+ + 219) 3:3 Vậy A chia hết cho 3 A = 2 + 2 ² + 2³ + 2ª +. 20 + 2.9+ +2 2+2 ³ + 2² +2²4 + + 218 + 720 +2²³ +2²+ +218 +220 2. (2 +2²) + 2² (1+2²) +.. + 218 ( 1 +2²) = 2 5 +2²5 + + 218 5. 12 +2° + 2 ... +218 ) 5 : 5. vậy A chia hết cho 5

Ta có : 

\(A=2+2^2+2^3+2^4...2^{2010}\)\(^0\)

\(=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{2009}\left(1+2\right)\)

\(=2.3+2^3.3+....+2^{2009}.3\)

\(=3\left(2+2^3+....+2^{2009}\right)⋮3\)

Ta có :

\(2+2^2+2^3+2^4+....+2^{2010}\)

\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+...+2^{2008}\left(1+2+2^2\right)\)

\(=2.7+2^4.7+....+2^{2008}.7\)

\(=7\left(2+2^4+....+2^{2008}\right)⋮7\)

Vậy \(2^1+2^2+2^3+2^4+...+2^{2010}⋮3\) và \(7\)

b) A=2+2²+2³+...+2²⁰

A=(2+2²)+(2³+2⁴)+...+(2¹⁹+2²⁰)

A=3.2+3.2³+...+3.2¹⁹

A=3.(2+2³+2⁵+...+2¹⁹)

⇒A⋮3

phần c) làm tương tự

Lời giải:
$A=(2+2^2)+(2^3+2^4)+....+(2^{99}+2^{100})$
$=2(1+2)+2^3(1+2)+...+2^{99}(1+2)$

$=2.3+2^3.3+...+2^{99}.3$

$=3(2+2^3+...+2^{99})\vdots 3$

Ta có đpcm.

a) \(A=2+2^2+2^3+...+2^{20}\)

\(A=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{19}+2^{20}\right)\)

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

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

Vậy A chia hết cho 3

________

\(A=2+2^2+2^3+...+2^{20}\)

\(A=\left(2+2^3\right)+\left(2^2+2^4\right)+...+\left(2^{58}+2^{60}\right)\)

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

\(A=5\cdot\left(2+2^2+...+2^{58}\right)\)

Vậy A chia hết cho 5 

1: \(A=2+2^2+2^3+2^4+...+2^{97}+2^{98}+2^{99}+2^{100}\)

\(=2\left(1+2+2^2+2^3\right)+...+2^{97}\left(1+2+2^2+2^3\right)\)

\(=15\left(2+2^5+...+2^{97}\right)\)

\(=30\left(1+2^4+...+2^{96}\right)⋮30\)

2:

\(B=3+3^2+3^3+...+3^{2022}\)

\(=\left(3+3^2\right)+\left(3^3+3^4\right)+...+\left(3^{2021}+3^{2022}\right)\)

\(=\left(3+3^2\right)+3^2\left(3+3^2\right)+...+3^{2020}\left(3+3^2\right)\)

\(=12\left(1+3^2+...+3^{2020}\right)⋮12\)

 

A=\((1+2)+\left(2^2+2^3\right)+...+\left(2^{19}+2^{20}\right)\)

A=\(3.1+2^2\left(1+2\right)+...+2^{19}\left(1+2\right)\)

A=\(3.1+3.2^2+...+3.2^{19}\)

A=\(3\left(1+2^2+...+2^{19}\right)\)\(⋮3\)

Vậy A\(⋮3\)

A=(1+2)+(22+23)+...+(219+220)(1+2)+(22+23)+...+(219+220)

A=3.1+22(1+2)+...+219(1+2)3.1+22(1+2)+...+219(1+2)

A=3.1+3.22+...+3.2193.1+3.22+...+3.219

A=3(1+22+...+219)3(1+22+...+219)⋮3⋮3

NÊN  A⋮3

Sửa đề: \(A=2+2^2+...+2^{60}\)

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

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

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

=>A chia hết cho 15 và A chia hết cho 3

\(A=2+2^2+...+2^{60}\)

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

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

\(=7\left(2+2^4+...+2^{58}\right)⋮7\)

Ta có: \(A⋮3;A⋮7\)

ƯCLN(3;7)=1

Do đó: \(A⋮3\cdot7=21\)