[Toán 7] CHứng minh S<18/19

M

maihuy409

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$\RightarrowS<$\frac{1}{1.2}+\frac{1}{2.3}+...+1/9.10
$\RightarrowS<$\frac{1}{1}-$\frac{1}{10}
$\RightarrowS<$\frac{9}{10}
$\RightarrowS<\frac{18}{19}
 
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K

khanhlinh2018

12+18+118...+1162\frac{1}{2}+\frac{1}{8}+\frac{1}{18}...+\frac{1}{162}
=12(112+122+132+...+192)=\frac{1}{2}(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{9^2})
<12(112−0,25+122−0,25+132−0,25+...+192−0,25)<\frac{1}{2}(\frac{1}{1^2-0,25}+\frac{1}{2^2-0,25}+\frac{1}{3^2-0,25}+...+\frac{1}{9^2-0,25})
=12(10,5.1,5+11,5.2,5+12,5.3,5+...+18,5.9,5)=\frac{1}{2}(\frac{1}{0,5.1,5}+\frac{1}{1,5.2,5}+\frac{1}{2,5.3,5}+...+\frac{1}{8,5.9,5})
=12(11,5−12,5+12,5−13,5+...+18,5−19,5)=\frac{1}{2}(\frac{1}{1,5}-\frac{1}{2,5}+\frac{1}{2,5}-\frac{1}{3,5}+...+\frac{1}{8,5}-\frac{1}{9,5})
=1819=\frac{18}{19}
 
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