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\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+.....+\frac{1}{2001^2}+\frac{1}{2002^2}\)
\(\Rightarrow A< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.......+\frac{1}{2000.2001}+\frac{1}{2001.2002}\)
\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.......+\frac{1}{2000}-\frac{1}{2001}+\frac{1}{2001}-\frac{1}{2002}\)
\(\Rightarrow A< 1-\frac{1}{2002}=\frac{2001}{2002}\left(đpcm\right)\)
Đặt A =\(\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+...+\frac{2008}{3^{2008}}\)
Suy ra 3A = \(1+\frac{2}{3}+\frac{3}{3^2}+...+\frac{2008}{3^{2007}}\)=> 2A = 3A - A = \(1+\frac{2}{3}+\frac{3}{3^2}+...+\frac{2008}{3^{2007}}-\frac{1}{3}-\frac{2}{3^2}-\frac{3}{3^3}-...-\frac{2008}{3^{3008}}\)= \(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2007}}-\frac{2008}{3^{2008}}\)
= \(\frac{3}{2}-\frac{1}{2.3^{2007}}\)Suy ra A = \(\frac{3}{4}-\frac{1}{8.3^{2007}}\)<\(\frac{3}{4}\)(ĐPCM)
Ta có : \(A=\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}+......+\frac{1}{2^{100}}\)
\(\Rightarrow4A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^4}+.....+\frac{1}{2^{98}}\)
\(\Rightarrow4A-A=\frac{1}{2}-\frac{1}{2^{100}}\)
\(\Rightarrow3A=\frac{2^{99}-1}{2^{100}}\)
\(\Rightarrow A=\frac{2^{99}-1}{\frac{2^{200}}{3}}\)
Vì : \(\frac{2^{99}-1}{2^{200}}< 1\)
Nên : \(A< \frac{1}{3}\)
Đặt \(A=\frac{1}{2^2}-\frac{1}{2^4}+\frac{1}{2^6}-...-\frac{1}{2^{2004}}\)
\(2^2A=1-\frac{1}{2^2}+\frac{1}{2^4}-...-\frac{1}{2^{2002}}\)
\(2^2A+A=\left(1-\frac{1}{2^2}+\frac{1}{2^4}-...-\frac{1}{2^{2002}}\right)+\left(\frac{1}{2^2}-\frac{1}{2^4}+\frac{1}{2^6}-...-\frac{1}{2^{2004}}\right)\)
\(5A=1-\frac{1}{2^{2004}}\)
\(\Rightarrow5A< 1\Rightarrow A< \frac{1}{5}\left(đpcm\right)\)
Đặt A = \(\frac{1}{2^2}-\frac{1}{2^4}+\frac{1}{2^6}-\frac{1}{2^8}+...+\frac{1}{2^{2002}}-\frac{1}{2^{2004}}\)
=> 22A = 4A = \(1-\frac{1}{2^2}+\frac{1}{2^4}-\frac{1}{2^6}+...+\frac{1}{2^{2000}}-\frac{1}{2^{2002}}\)
=> 4A + A =\(1-\frac{1}{2^2}+\frac{1}{2^4}-\frac{1}{2^6}+...+\frac{1}{2^{2000}}-\frac{1}{2^{2002}}+\frac{1}{2^2}-\frac{1}{2^4}+\frac{1}{2^6}-\frac{1}{2^8}+...+\frac{1}{2^{2002}}-\frac{1}{2^{2004}}\)
=> 5A = \(1-\frac{1}{2^{2004}}\)
=> \(A=\frac{1}{5}-\frac{1}{2^{2004}.5}< \frac{1}{5}=0,2\)
=> A < 0,2 (ĐPCM)
\(|x-\frac{1}{3}|=|\left(-3.2\right)+\frac{2}{5}|\)
\(\Rightarrow|x-\frac{1}{3}|=|-3.2+0.4|\)
\(\Rightarrow|x-\frac{1}{3}|=|-2.8|\)
\(\Rightarrow|x-\frac{1}{3}|=2.8\)
\(\Rightarrow\orbr{\begin{cases}x-\frac{1}{3}=2.8\\x-\frac{1}{3}=-2.8\end{cases}}\)\(\Rightarrow\orbr{\begin{cases}x=\frac{43}{15}\\x=-\frac{41}{15}\end{cases}}\)
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