Question

Rút gọn

2^100+2^99+2^98+...+2^2+2^1

Answer 1

Đặt A = 2 ^ 100 + 2 ^ 99 + 2 ^ 98 + ... + 2 ^ 2 + 2 ^ 1

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

2A - A = ( 2 ^ 101 + 2 ^ 100 + 2 ^ 99 + ... + 2 ^ 3 + 2 ^ 2 )

           -  ( 2 ^ 100 + 2 ^ 99 + 2 ^ 98 + ... + 2 ^ 2 + 2 ^ 1 )

 A        = 2 ^ 101 - 2

Answer 2

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

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

\(2A=2^{101}+2^{100}+...+2^2+2^1\)

\(A=2A-A\)

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

Answer 3

Viết biểu thức A thành:
\(A=\left(2^{100}+2^{98}+...+2^2\right)-\left(2^{99}-2^{97}-2\right)=M-N\)
Ta có \(M=2^{100}+2^{98}+...+2^2\)
\(\Rightarrow2^2M=2^{102}+2^{100}+...+2^4\)
\(\Rightarrow4M-M=2^{102}-2^2\Rightarrow M=\frac{3^{102}-2}{3}\)
Tương tự với \(N=2^{99}-2^{97}-...-2=\frac{3^{101}-3^{100}+2}{3}\)
NHư vậy \(M-N=\frac{2^{102}-2-2^{101}+2^{100}-2}{3}=\frac{3^{101}.34}{3}\)