a, \(B=\dfrac{3}{1.4}+\dfrac{3}{4.7}+...+\dfrac{3}{27.30}.\)
\(=1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+...+\dfrac{1}{27}-\dfrac{1}{30}.\)
\(=1+\left(\dfrac{1}{4}-\dfrac{1}{4}\right)+\left(\dfrac{1}{7}+\dfrac{1}{7}\right)+...+\left(\dfrac{1}{27}-\dfrac{1}{27}\right)-\dfrac{1}{30}.\)
\(=1+0+0+...+0-\dfrac{1}{30}.\)
\(=1-\dfrac{1}{30}=\dfrac{29}{30}.\)
Vậy \(B=\dfrac{29}{30}.\)
b, \(B=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dfrac{1}{5^2}+\dfrac{1}{6^2}+\dfrac{1}{7^2}+\dfrac{1}{8^2}.\)
\(=\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+\dfrac{1}{5.5}+\dfrac{1}{6.6}+\dfrac{1}{7.7}+\dfrac{1}{8.8}.\)
\(< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{6.7}+\dfrac{1}{7.8}.\)
\(< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{8}.\)
\(< 1+\left(\dfrac{1}{2}-\dfrac{1}{2}\right)+\left(\dfrac{1}{3}-\dfrac{1}{3}\right)+\left(\dfrac{1}{4}-\dfrac{1}{4}\right)+\left(\dfrac{1}{5}-\dfrac{1}{5}\right)+\left(\dfrac{1}{6}-\dfrac{1}{6}\right)+\left(\dfrac{1}{7}-\dfrac{1}{7}\right)-\dfrac{1}{8}.\)
\(< 1+0+0+0+0+0+0-\dfrac{1}{8}.\)
\(< 1-\dfrac{1}{8}=\dfrac{7}{8}< 1.\)
\(\Rightarrow B< \dfrac{7}{8}< 1.\)
\(\Rightarrow B< 1\left(đpcm\right).\)
~ Chúc bn học tốt!!! ~
Bài mik đúng thì nhớ tick mik nha!!! ^ - ^
Câu 5:
a) Dễ lắm!!!
Ta có: A = \(\left(1-\dfrac{1}{2010}\right)\) . \(\left(1-\dfrac{2}{2010}\right)\) .... \(\left(1-\dfrac{2011}{2010}\right)\)
=> A = \(\left(1-\dfrac{1}{2010}\right)\). \(\left(1-\dfrac{2}{2010}\right)\) .....\(\left(1-\dfrac{2010}{2010}\right)\)\(\left(1-\dfrac{2011}{2010}\right)\) ( ghi hàng này cho em hiểu thôi chứ ko cần)
=> A = \(\left(1-\dfrac{1}{2010}\right)\) ......(1-1)\(\left(1-\dfrac{2011}{2010}\right)\)
=> A = \(\left(1-\dfrac{1}{2010}\right)\) .....0. \(\left(1-\dfrac{2011}{2010}\right)\) = 0
b) Ta có: A = \(\dfrac{1946}{1986}\) = \(1-\dfrac{40}{1986}\)
B = \(\dfrac{1968}{2008}\) = \(1-\dfrac{40}{2008}\)
Vì \(\dfrac{40}{1986}\) > \(\dfrac{40}{2008}\) ( Do 1968 < 2008)
=> \(1-\dfrac{40}{1986}\) < \(1-\dfrac{40}{2008}\)
=> A < B
P/s: Hơi muộn nhỉ!!!! ( do bây giờ mới rảnh để trả lời )
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