Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
\(A=\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{3.5}-\frac{1}{5.7}+...+\frac{1}{1997.1999}-\frac{1}{1999.2001}\)
\(=\frac{1}{1.3}-\frac{1}{1999.2001}\)
Bạn tính kết quả nhé
\(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{1}{2014}}\)
Xét mẫu:
\(\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{1}{2014}\)
= \(\left(1+\frac{2013}{2}\right)+\left(1+\frac{2012}{3}\right)+...+\left(1+\frac{1}{2014}\right)+1\)
= \(\frac{2014}{2}+\frac{2014}{3}+....+\frac{2014}{2013}+\frac{2014}{2014}\)
= \(2014\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}\right)\)
\(\Rightarrow S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{2014.\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}\right)}\)
\(\Rightarrow S=\frac{1}{2014}\)
Nhân 2 vế của 2 ĐT đề bài ta có
\(\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}\right)=\frac{47}{10}\)
<=> \(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+\left(\frac{a}{a+b}+\frac{b}{a+b}\right)+\left(\frac{b}{b+c}+\frac{c}{b+c}\right)+\left(\frac{c}{a+c}+\frac{a}{a+c}\right)=\frac{47}{10}\)
=>\(P=\frac{17}{10}\)
Vậy \(P=\frac{17}{10}\)
1)\(S=\frac{1}{1.3.5}+\frac{1}{3.5.7}+\frac{1}{5.7.9}+...+\frac{1}{2003.2005.2007}\)
\(\Rightarrow4S=\frac{4}{1.3.5}+\frac{4}{3.5.7}+\frac{4}{5.7.9}+...+\frac{4}{2003.2005.2007}\)
\(=\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{3.5}-\frac{1}{5.7}+\frac{1}{5.7}-\frac{1}{7.9}+...+\frac{1}{2003.2005}-\frac{1}{2005.2007}\)
\(=\frac{1}{3}-\frac{1}{4024035}=\frac{1341345}{4024035}=\frac{1}{3}\)
\(\Rightarrow S=\frac{1}{3}:4\approx0,08\)
2)\(S=\frac{1}{3}:4=\frac{1}{12}\)
\(S=\frac{1}{1.3.5}+\frac{1}{3.5.7}+\frac{1}{5.7.9}+...+\frac{1}{2003.2005.2007}\)
\(S=\frac{2}{2}.\frac{1}{1.3.5}+\frac{2}{2}.\frac{1}{3.5.7}+\frac{2}{2}.\frac{1}{5.7.9}+...+\frac{2}{2}.\frac{1}{2003.2005.2007}\)
\(S=\frac{1}{2}.\frac{2}{1.3.5}+\frac{1}{2}.\frac{2}{3.5.7}+\frac{1}{2}.\frac{2}{5.7.9}+...+\frac{1}{2}.\frac{2}{2003.2005.2007}\)
\(S=\frac{1}{2}.\left(\frac{2}{1.3.5}+\frac{2}{3.5.7}+\frac{2}{5.7.9}+...+\frac{2}{2003.2005.2007}\right)\)
\(S=\frac{1}{2}.\left(\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{3.5}-\frac{1}{5.7}+\frac{1}{5.7}-\frac{1}{7.9}+...+\frac{1}{2003.2005}-\frac{1}{2005.2007}\right)\)
\(S=\frac{1}{2}\left(\frac{1}{1.3}-\frac{1}{2005.2007}\right)=\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{4024035}\right)=\frac{1}{2}.\frac{1341345}{4024035}=\frac{1}{2}.\frac{1}{3}=\frac{1}{6}\)
Vậy \(S=\frac{1}{6}\)