Câu 1:
\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)..\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)\)
\(A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{2015}{2016}.\frac{2016}{2017}\)
\(A=\frac{1}{2017}\)
Vậy ..............................
Phần giống nhau là gạch ý!
Câu 2
\(S=2^{2010}-2^{2009}-2^{2008}-...-2-1\)
\(\Rightarrow S=2^{2010}-\left(2^{2009}+2^{2008}+...+2+1\right)\)
Đặt \(Q=2^{2009}+2^{2008}+...+2+1\)
\(\Rightarrow2Q=2^{2010}+2^{2009}+...+4+2\)
\(\Rightarrow2Q-Q=\left(2^{2010}+2^{2009}+...+4+2\right)-\left(2^{2009}+2^{2008}+...+2+1\right)\)
\(\Rightarrow Q=2^{2010}-1\)
\(\Rightarrow S=2^{2010}-\left(2^{2010}-1\right)\)
\(\Rightarrow S=2^{2010}-2^{2010}+1\)
\(\Rightarrow S=1\)
Vậy .........................
b) \(S=2^{2010}-2^{2009}-2^{2008}-...-2-1\)
\(\Rightarrow2S=2^{2011}-2^{2010}-2^{2009}-...-2^2-2\)
\(\Rightarrow2S-S=\left(2^{2011}-2^{2010}-2^{2009}-...-2^2-2\right)-\left(2^{2010}-2^{2009}-2^{2008}-...-2-1\right)\)
\(\Rightarrow S=2^{2011}-2^{2010}-2^{2009}-...-2^2-2-2^{2010}+2^{2009}+2^{2008}+...+2+1\)
\(\Rightarrow S=2^{2011}-2^{2010}-2^{2010}+1\)
\(\Rightarrow S=2^{2011}-2.2^{2010}+1\)
\(\Rightarrow S=2^{2011}-2^{2011}+1\)
\(\Rightarrow S=0+1\)
\(\Rightarrow S=1.\)
Vậy \(S=1.\)
Chúc bạn học tốt!