Ta có :
\(M=2^{2017}-\left(2^{2016}+2^{2017}+...............+2+1\right)\)
Đặt :
\(A=2^{2016}+2^{2015}+................+2+1\)
\(\Leftrightarrow2A=2^{2017}+2^{2016}+2^{2015}+............+2^2+2\)
\(\Leftrightarrow2A-A=\left(2^{2017}+2^{2016}+........+2\right)-\left(2^{2016}+2^{2015}+..........+1\right)\)
\(\Leftrightarrow A=2^{2017}-1\)
\(\Leftrightarrow M=2^{2017}-A\)
\(\Leftrightarrow M=2^{2017}-\left(2^{2017}-1\right)\)
\(\Leftrightarrow M=2^{2017}-2^{2017}+1\)
\(\Leftrightarrow M=0+1=1\)
\(M=2^{2017}-\left(2^{2016}+2^{2015}+...+2^1+2^0\right)\)
Đặt :
\(S=2^{2016}+2^{2015}+...+2^1+2^0\)
\(\Rightarrow S=2^0+2^1+...+2^{2015}+2^{2016}\)
\(\Rightarrow2S=2\left(2^0+2^1+...+2^{2015}+2^{2016}\right)\)
\(\Rightarrow2S=2^1+2^2+...+2^{2016}+2^{2017}\)
\(\Rightarrow2S-S=\left(2^1+2^2+...+2^{2016}+2^{2017}\right)-\left(2^0+2^1+...+2^{2015}+2^{2016}\right)\)
\(\Rightarrow S=2^{2017}-1\)
Thay S vào M ta có:
\(M=2^{2017}-\left(2^{2017}-1\right)\)
\(M=2^{2017}-2^{2017}+1=1\)