Đặt \(N=2^0+2^1+...+2^{2008}+2^{2009}\)

Suy ra: \(M=2^{2010}-N\)

Ta có: \(N=2^0+2^1+...+2^{2008}+2^{2009}\)

\(\Leftrightarrow2N=2+2^2+...+2^{2009}+2^{2010}\)

\(\Leftrightarrow N=2^{2010}-1\)

\(M=2^{2010}-N=2^{2010}-2^{2010}+1=1\)

\(M=2^{2010}-\left(2^{2009}+2^{2008}+...+2^1+2^0\right)\)

Gọi \(N=2^{2009}+2^{2008}+...+2^1+2^0\)

\(2N=2^{2010}+2^{2009}+...+2^2+2^1\\ 2N-N=\left(2^{2010}+2^{2009}+...+2^2+2^1\right)-\left(2^{2009}+2^{2008}+...+2^1+2^0\right)\\ N=2^{2010}-2^0\\ N=2^{2010}-1\)

Thay vào ta được

\(M=2^{2010}-\left(2^{2010}-1\right)\\ M=2^{2010}-2^{2010}+1\\ M=1\)

Vậy \(M=1\)

Ta có :

\(M=2^{2010}-\left(2^{2009}+2^{2008}+...+2^0\right)\)

Đặt A=2²⁰⁰⁹+2²⁰⁰⁸+..+2⁰

\(A=2^{2009}+2^{2008}+...+2^0\\ 2A=2^{2010}+2^{2009}+...+2^1\\ \Rightarrow2A-A=A=2^{2010}-2^0\\ \Rightarrow M=2^{2010}-\left(2^{2010}-2^0\right)\\ M=2^{2010}-2^{2010}+1\\ \Rightarrow M=1\)

Chúc bạn học tốt!

Đặt \(A=2^{2009}+2^{2008}+2^{2007}+...+2+1\\ \Rightarrow2A=2^{2010}+2^{2009}+2^{2008}+...+2^2+2\\ \Rightarrow2A-A=\left(2^{2010}+2^{2009}+2^{2008}+...+2^2+2\right)-\left(2^{2009}+2^{2008}+2^{2007}+...+2+1\right)\\ \Rightarrow A=2^{2010}-1\)

\(\Rightarrow M=2^{2010}-2^{2010}+1=1\)

Đặt A = 2²⁰⁰⁹ + 2²⁰⁰⁸ + ... + 2¹ + 2⁰

2A = 2²⁰¹⁰ + 2²⁰⁰⁹ + ... + 2² + 2¹

2A - A = (2²⁰¹⁰ + 2²⁰⁰⁹ + ... + 2² + 2¹) - (2²⁰⁰⁹ + 2²⁰⁰⁸ + ... + 2¹ + 2⁰)

A = 2²⁰¹⁰ - 2⁰

 A = 2²⁰¹⁰ - 1

=> 2²⁰¹⁰ - (2²⁰⁰⁹ + 2²⁰⁰⁸ + ... + 2¹ + 2⁰)

= 2²⁰¹⁰ - (2²⁰¹⁰ - 1)

= 2²⁰¹⁰ - 2²⁰¹⁰ + 1

= 1

Đặt A = 2²⁰⁰⁹ + 2²⁰⁰⁸ + ... + 2¹ + 2⁰

2A = 2²⁰¹⁰ + 2²⁰⁰⁹ + ... + 2² + 2¹

2A - A = (2²⁰¹⁰ + 2²⁰⁰⁹ + ... + 2² + 2¹) - (2²⁰⁰⁹ + 2²⁰⁰⁸ + ... + 2¹ + 2⁰)

A = 2²⁰¹⁰ - 2⁰

 A = 2²⁰¹⁰ - 1

=> 2²⁰¹⁰ - (2²⁰⁰⁹ + 2²⁰⁰⁸ + ... + 2¹ + 2⁰)

= 2²⁰¹⁰ - (2²⁰¹⁰ - 1)

= 2²⁰¹⁰ - 2²⁰¹⁰ + 1

= 1

\(M=2^{2010}-\left(2^{2009}+2^{2008}+....+2^1+2^0\right)\)

Đặt:

\(S=2^{2009}+2^{2008}+....+2^1+2^0\)

\(S=2^0+2^1+.....+2^{2008}+2^{2009}\)

\(2S=2\left(2^0+2^1+.....+2^{2008}+2^{2009}\right)\)

\(2S=2^1+2^2+.....+2^{2009}+2^{2010}\)

\(2S-S=\left(2^1+2^2+.....+2^{2009}+2^{2010}\right)-\left(2^0+2^1+.....+2^{2008}+2^{2009}\right)\)

\(S=2^{2010}-2^0=2^{2010}-1\)

Thay S vào M ta có:

\(M=2^{2010}-\left(2^{2010}-1\right)\)

\(M=2^{2010}-2^{2010}+1=1\)

M=2²⁰¹⁰-(2²⁰⁰⁹+2²⁰⁰⁸+2²⁰⁰⁷+...+2¹+2⁰)

M=2²⁰¹⁰-2²⁰⁰⁹-2²⁰⁰⁸-2²⁰⁰⁷-...-2¹-2⁰

=>2M=2²⁰¹¹-2²⁰¹⁰-2²⁰⁰⁹-2²⁰⁰⁸-...-2²-2¹

=>2M-M=2²⁰¹¹-2²⁰¹⁰-2²⁰⁰⁹-2²⁰⁰⁸-...-2²-2¹-(2²⁰¹⁰-2²⁰⁰⁹-2²⁰⁰⁸-2²⁰⁰⁷-...-2¹-2⁰)

=>M=2²⁰¹¹-2²⁰¹⁰-2²⁰⁰⁹-2²⁰⁰⁸-...-2²-2¹-2²⁰¹⁰+2²⁰⁰⁹+2²⁰⁰⁸+2²⁰⁰⁷+...+2¹+2⁰

=2²⁰¹¹-2²⁰¹⁰-2²⁰¹⁰+2⁰

=2²⁰¹¹-2.2²⁰¹⁰+1

=2²⁰¹¹-2²⁰¹¹+1

=1

Vậy M=1