\(A=2^{100}-2^{99}+2^{98}-2^{97}+....-2^3+2^2-2+1\\ A=\left(2^{100}+2^{98}+...+2\right)-\left(2^{99}+2^{97}+...+1\right)\)

Gọi \(\left(2^{100}+2^{98}+...+2\right)\)là B

\(B=\left(2^{100}+2^{98}+...+2\right)\\ 2B=2^{102}+2^{100}+.....+2^2\\ 2B-B=\left(2^{102}+2^{100}+.....+2^2\right)-\left(2^{100}+2^{98}+...+2\right)\\ B=2^{102}-2\)

Gọi \(\left(2^{99}+2^{97}+...+1\right)\) là C

\(C=\left(2^{99}+2^{97}+...+1\right)\\ 2C=2^{101}+2^{99}+....+2\\ 2C-C=\left(2^{101}+2^{99}+9^{97}+...+2\right)-\left(2^{99}+9^{97}+...+1\right)\\ C=2^{101}-1\)

\(A=B+C\\ =>A=2^{102}-2+2^{101}-1\\ A=2^{101}\left(2+1\right)-3\\ A=2^{101}\cdot3-3\\ A=3\cdot\left(2^{101}-1\right)\)

\(\dfrac{1}{2}A=2^{99}-2^{98}+...-1+\dfrac{1}{2}\\ \Rightarrow A-\dfrac{1}{2}A=2^{100}-\dfrac{1}{2}\\ \Rightarrow A=2^{101}-1\)

giúp minh

\(2^{100}-2^{99}+2^{98}-2^{97}+2^{96}-2^{95}+...+2^4-2^3+2^2\)

\(=\left(2^{100}-2^{99}+2^{98}\right)-\left(2^{97}-2^{96}+2^{95}\right)+...+\left(2^4-2^3+2^2\right)\)

\(=2^{96}\left(2^4-2^3+2^2\right)-2^{93}\left(2^4-2^3+2^2\right)+...+\left(2^4-2^3+2^2\right)\)

\(=12\left(2^{96}-2^{93}+...+1\right)⋮12\)

\(A=2^{100}-\left(2^{99}+2^{98}+...+2+1\right)\)

Đặt \(B=2^{99}+2^{98}+...+2+1\)

\(\Rightarrow2B=2^{100}+2^{99}+...+2^2+2\)

\(\Rightarrow2B-B=2^{100}-1\Leftrightarrow B=2^{100}-1\)

\(\Rightarrow A=2^{100}-\left(2^{100}-1\right)=1\)

a) \(A=1+2+2^2+...+2^{50}\)

\(\Rightarrow2A=2+2^2+...+2^{51}\)

\(\Rightarrow A=2A-A=2+2^2+...+2^{51}-1-2-2^2-...-2^{50}=2^{51}-1\)

b) \(B=1+3+3^2+...+3^{100}\)

\(\Rightarrow3B=3+3^2+...+3^{101}\)

\(\Rightarrow2B=3B-B=3+3^2+...+3^{101}-1-3-3^2-...-3^{100}=3^{101}-1\)

\(\Rightarrow B=\dfrac{3^{101}-1}{2}\)

c) \(C=5+5^2+...+5^{30}\)

\(\Rightarrow5C=5^2+5^3+...+5^{31}\)

\(\Rightarrow4C=5C-C=5^2+5^3+...+5^{31}-5-5^2-...-5^{30}=5^{31}-5\)

\(\Rightarrow C=\dfrac{5^{31}-5}{4}\)

d) \(D=2^{100}-2^{99}+2^{98}-...+2^2-2\)

\(\Rightarrow2D=2^{101}-2^{100}+2^{99}-...+2^3-2^2\)

\(\Rightarrow3D=2D+D=2^{101}-2^{100}+2^{99}-...+2^3-2^2+2^{100}-2^{99}+...+2^2-2=2^{101}-2\)

\(\Rightarrow D=\dfrac{2^{101}-2}{3}\)

\(\Leftrightarrow2P=2^2+2^3+...+2^{100}\\ \Leftrightarrow2P-P=2^2+2^3+...+2^{100}-2-2^2-...-2^{99}\\ \Leftrightarrow P=2^{100}-2\)

Tính giá trị của biểu thức:

a) (27 + 86) - (29 - 5 + 84);

=  27 + 86 – 29 + 5 - 84

= 5

b) 39 - (298 - 89) + 299.

= 39 – 298 + 89 + 299

= 129

) (27 + 86) - (29 - 5 + 84) 

= 27 + 86 - 29 + 5 - 84 

= (27 – 29) + (86 – 84) + 5

= (- 2) + 2 + 5

= 0 + 5

= 5

b) 39 - (298 - 89) + 299 

= 39 - 298 + 89 + 299

= (39 + 89) + (299 – 298)

= 39 + 89 + 1

= 39 + (89 + 1)

= 39 + 90

= 129

Theo bài ra, ta chia thành các nhóm như sau:

1+(2-3-4+5)+(6-7-8+9)+.......+(298-299-300+301)+302 = 1+0+0+0+0+.......+0+302 = 1+302=303

                  Đáp số: 303

D/s303

303 nha bn 

chúc linh học giỏi