\(S=1+2^2+2^4+2^6+...+2^{100}\)

\(2^2S=2^2\left(1+2^2+2^4+2^6+...+2^{100}\right)\)

\(4S=2^2+2^4+2^6+2^8+...+2^{102}\)

\(4S-S=\left(2^2+2^4+2^6+2^8+...+2^{102}\right)-\left(1+2^2+2^4+2^6+...+2^{100}\right)\)

\(3S=2^{102}-1\)

\(S=\dfrac{2^{102}-1}{3}\)

`#3107.101107`

Gọi biểu thức trên là A

Ta có:

\(A=1+5^2+5^4+...+5^{40}\\ =1\cdot\left(1+5^2\right)+5^4\cdot\left(1+5^2\right)+...+5^{38}\cdot\left(1+5^2\right)\\ =\left(1+5^2\right)\cdot\left(1+5^4+...+5^{38}\right)\\ =26\cdot\left(1+5^4+...+5^{38}\right)\)

Vì \(26\cdot\left(1+5^4+...+5^{38}\right)\text{ }⋮\text{ }26\)

\(\Rightarrow A\text{ }⋮\text{ }26\)

_______

Gọi biểu thức trên là B

Ta có:

\(B=1+2^2+2^4+...+2^{100}\\ =1\cdot\left(1+2^2+2^4\right)+2^6\cdot\left(1+2^2+2^4\right)+...+2^{96}\cdot\left(1+2^2+2^4\right)\\ =\left(1+2^2+2^4\right)\cdot\left(1+2^6+...+2^{96}\right)\\ =21\cdot\left(1+2^6+...+2^{96}\right)\)

Vì \(21\cdot\left(1+2^6+...+2^{96}\right)\text{ }⋮\text{ }21\)

\(\Rightarrow B\text{ }⋮\text{ }21\)

_______

Gọi biểu thức trên là C

Ta có:

\(C=1+3^2+3^4+...+3^{100}\\ =1\cdot\left(1+3^2+3^4+3^6\right)+3^6\cdot\left(1+3^2+3^4+3^6\right)+...+3^{94}\cdot\left(1+3^2+3^4+3^6\right)\\ =\left(1+3^2+3^4+3^6\right)\cdot\left(1+3^6+...+3^{94}\right)\\ =820\cdot\left(1+3^6+...+3^{94}\right)\)

Vì \(820\cdot\left(1+3^6+...+3^{94}\right)\text{ }⋮\text{ }82\)

\(\Rightarrow C\text{ }⋮\text{ }82.\)

a) \(A=1+5^2+5^4+5^6...+5^{40}\)

\(\Rightarrow A=\left(1+5^2\right)+5^4\left(1+5^2\right)+...+5^{38}\left(1+5^2\right)\)

\(\Rightarrow A=26+5^4.26+...+5^{38}.26\)

\(\Rightarrow A=26\left(1+5^4+...+5^{38}\right)⋮26\)

\(\Rightarrow1+5^2+5^4+5^6...+5^{40}⋮6\left(dpcm\right)\)

b) \(B=1+2^2+2^4+2^6+...+2^{100}\)

\(\Rightarrow B=\left(1+2^2+2^4\right)+2^6\left(1+2^2+2^4\right)+...+2^{96}\left(1+2^2+2^4\right)\)

\(\Rightarrow B=21+2^6.21+...+2^{96}.21\)

\(\Rightarrow B=21\left(1+2^6+...+2^{96}\right)⋮21\)

\(\Rightarrow1+2^2+2^4+2^6+...+2^{100}⋮21\left(dpcm\right)\)

Bài C tương tự bạn tự làm nhé!

Có : \(S=1+2+2^2+2^3+....+2^{99}\)

\(\Rightarrow2S=2+2^2+2^3+....+2^{100}\)

\(\Rightarrow2S-S=\left(2+2^2+2^3+...+2^{100}\right)-\left(1+2+2^2+....+2^{99}\right)\)

\(\Rightarrow S=2^{100}-1< 2^{100}\)

Vậy \(S< 2^{100}\)

S=2¹⁰⁰-1

vì 2¹⁰⁰ -1<2¹⁰⁰

⇒S<2¹⁰⁰

ĐỀ CÓ LỖI KO BNJ ? 

\(A=2+2^2+2^3+2^4+...+2^{99}+2^{100}\)

\(\Rightarrow2A=2^2+2^3+2^4+...+2^{100}+2^{101}\)

\(\Rightarrow A=2A-A=2^2+2^3+2^4+...+2^{100}+2^{101}-2-2^2-2^3-2^4-...-2^{99}-2^{100}=2^{101}-2\)

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

\(\Rightarrow2A=2+2^2+2^3+2^4+2^5+...+2^{100}+2^{101}\)

\(\Rightarrow2A-A=2^{101}-1\)

\(\Leftrightarrow A=2^{101}-1\)

Đặt biểu thức là A

ta có 2A-A=2^101-1

a, Ta có :

 A =  1 + 2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100

2A =  2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100 + 2 101

A = 2A – A =  ( 2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100 + 2 101 ) –( 1 + 2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100 )

=  2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100 + 2 101 – 1 - 2 - 2 2 - 2 3 - 2 4 - . . . - 2 99 - 2 100

=  2 101 - 1

Vậy A =  2 101 - 1

b, Ta có.

B = 5 + 5 3 + 5 5 + . . . + 5 97 + 5 99

5 2 B =  5 2 ( 5 + 5 3 + 5 5 + . . . + 5 97 + 5 99 )

25B =  5 3 + 5 5 + . . . + 5 97 + 5 99 + 5 101

25B – B = ( 5 3 + 5 5 + . . . + 5 97 + 5 99 + 5 101 ) –  ( 5 + 5 3 + 5 5 + . . . + 5 97 + 5 99 )

24B =  5 3 + 5 5 + . . . + 5 97 + 5 99 + 5 101 – 5 - 5 3 - 5 5 - . . . - 5 97 - 5 99

24B =  5 101 - 5

B =  5 101 - 5 24 = 5 5 100 - 1 24

Vậy B =  5 5 100 - 1 24

Tính các tổng sau:

1, S=1-2+3_4+..+25-26

S =-1+3-5+7-...-53+55                       ( có 28 số hạng )

   = (-1+3)+(-5+7)+...+(-53+55)         ( có 28:2=14 nhóm )

   = 2+2+...+2

    = 2 . 14

     = 28

Á nhầm rùi xl bn nha

S = 22 + 24 + 26 + ... + 346

= (346 + 22).[(346 - 22) : 2 + 1] : 2

= 29992