Cho A =\(\left(\frac{1}{4}-1\right).\left(\frac{1}{9}-1\right).\left(\frac{1}{16}-1\right)...\left(\frac{1}{400}-1\right)\)so sánh A với\(-\frac{1}{2}\)
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3 tháng 5 2018
\(A=\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)\left(\frac{1}{16}-1\right)...\left(\frac{1}{400}-1\right)\)
\(-A=\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)...\left(1-\frac{1}{400}\right)\)
\(-A=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{399}{400}\)
\(-A=\frac{1\cdot3}{2\cdot2}\cdot\frac{2.4}{3.3}\cdot\frac{3.5}{4.4}\cdot...\cdot\frac{19.21}{20.20}\)
\(-A=\frac{1\cdot2\cdot3\cdot...\cdot19}{2\cdot3\cdot4\cdot...\cdot20}\cdot\frac{3\cdot4\cdot5\cdot...\cdot21}{2\cdot3\cdot4\cdot...\cdot20}\)
\(-A=\frac{1}{20}\cdot\frac{21}{2}=\frac{21}{40}>\frac{20}{40}=\frac{1}{2}\)
\(-A>\frac{1}{2}\Rightarrow A< \frac{1}{2}\)
23 tháng 8 2018
Ta có : \(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)...\left(1-\frac{1}{19}\right)\left(1-\frac{1}{20}\right)\)
\(=\frac{1}{2}.\frac{2}{3}....\frac{18}{19}.\frac{19}{20}\)
\(=\frac{1.2....18.19}{2.3...19.20}\)
\(=\frac{1}{20}>\frac{1}{21}\)
Vậy A > 1/21
VC
Vũ Cao Minh⁀ᶦᵈᵒᶫ ( ✎﹏IDΣΛ亗 )
CTV
VIP
2 tháng 8 2019
\(\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}.....\frac{899}{30^2}\)
\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{29.31}{30.30}=\frac{1.2.3.....29}{2.3.4.....30}.\frac{3.4.5.....31}{2.3.4.....30}\)
\(=\frac{1}{2}.\frac{31}{30}=\frac{31}{60}\)
26 tháng 7 2017
\(B=\left(1-\frac{1}{4}\right).\left(1-\frac{1}{9}\right).\left(1-\frac{1}{16}\right)...\left(1-\frac{1}{81}\right).\left(1-\frac{1}{100}\right)\)
\(B=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}...\frac{80}{81}.\frac{99}{100}\)
\(B=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}...\frac{8.10}{9.9}.\frac{9.11}{10.10}\)
\(B=\frac{1.2.3...8.9}{2.3.4...9.10}.\frac{3.4.5...10.11}{2.3.4...9.10}\)
\(B=\frac{1}{10}.\frac{11}{2}\)
\(B=\frac{11}{20}>\frac{11}{21}\)
TM
1 tháng 10 2016
\(A=\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)...\left(\frac{1}{10}-1\right)\)
\(\Rightarrow A=\frac{-1}{2}.\frac{-2}{3}.\frac{-3}{4}...\frac{-9}{10}\)
\(\Rightarrow A=\frac{-1}{10}\)
Dễ thấy \(\frac{1}{10}< \frac{1}{9}\Rightarrow\frac{-1}{10}>\frac{-1}{9}\)
14 tháng 8 2017
\(A=\frac{-1}{2}.\frac{-2}{3}.\frac{-3}{4}.....\frac{-9}{10}\)
\(A=\frac{-1}{10}\)
\(\frac{-1}{10}>\frac{-1}{9}\Rightarrow A>\frac{-1}{9}\)
đ/s:..
A=\(\left(\frac{1}{4}-1\right).\left(\frac{1}{9}-1\right).\left(\frac{1}{16}-1\right).\left(\frac{1}{25}-1\right).....\left(\frac{1}{361}-1\right).\left(\frac{1}{400}-1\right)\)
A=\(\left(-\frac{3}{4}\right).\left(-\frac{8}{9}\right).\left(-\frac{15}{16}\right).\left(-\frac{24}{25}\right).....\left(-\frac{360}{361}\right).\left(-\frac{399}{400}\right)\)
A=\(\left(-\frac{1.3}{2.2}\right).\left(-\frac{2.4}{3.3}\right).\left(-\frac{3.5}{4.4}\right).\left(-\frac{4.6}{5.5}\right).....\left(-\frac{18.20}{19.19}\right).\left(-\frac{19.21}{20.20}\right)\)
A=\(\left(-\frac{1}{2}\right).\left(-\frac{1}{1}\right).\left(-\frac{1}{1}\right).\left(-\frac{1}{1}\right).....\left(-\frac{1}{1}\right).\left(-\frac{21}{20}\right)\)
A=\(-\frac{1}{2}.\frac{21}{20}=-\frac{21}{40}< -\frac{1}{2}\)
\(\Rightarrow A< -\frac{1}{2}\)