\(S=2014+\frac{2014}{1+2}+\frac{2014}{1+2+3}+...+\frac{2014}{1+2+3+...+10000}\)

\(S=\frac{2014}{\frac{1.2}{2}}+\frac{2014}{\frac{2.3}{2}}+\frac{2014}{\frac{3.4}{2}}+...+\frac{2014}{\frac{10000.10001}{2}}\)

\(S=\frac{4028}{1.2}+\frac{4028}{2.3}+\frac{4028}{3.4}+...+\frac{4028}{10000.10001}\)

\(S=4028\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{10000.10001}\right)\)

\(S=4028\left(\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{10001-10000}{10000.10001}\right)\)

\(S=4028\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{10000}-\frac{1}{10001}\right)\)

\(S=4028\left(1-\frac{1}{10001}\right)=\frac{40280000}{10001}\)

\(a_{n-1}=\frac{2}{n\left(n+1\right)}=\frac{2}{n}+\frac{2}{n+1}\)

\(A=\frac{2}{2}-\frac{2}{3}+\frac{2}{3}-\frac{2}{4}+\frac{2}{4}-\frac{2}{5}+.......+\frac{2}{2014}-\frac{2}{2015}=1-\frac{2}{2015}=\frac{2013}{2015}\)

rảnh quá ngồi bấm, nếu bấm máy tính thì tự ngồi tạo công thức chứ rảnh ghê

\(A=\left(\frac{1}{1+2}\right).\left(\frac{1}{1+2+3}\right).....\left(\frac{1}{1+2+3+...+2014}\right)\)

\(A=\left(\frac{1}{\frac{2.\left(2+1\right)}{2}}\right).\left(\frac{1}{\frac{3.\left(3+1\right)}{2}}\right).....\left(\frac{1}{\frac{2014.\left(2014+1\right)}{2}}\right)\)

\(A=\frac{1}{\frac{2.3}{2}}.\frac{1}{\frac{3.4}{2}}.\frac{1}{\frac{4.5}{2}}.....\frac{1}{\frac{2014.2015}{2}}\)

\(A=\frac{2}{2.3}.\frac{2}{3.4}.\frac{2}{4.5}.....\frac{2}{2014.2015}\)

Đến đây thì không tính được nữa , có thể bạn chép nhầm dấu cộng thành dấu nhân rồi.

Nếu đổi dấu nhân thành dấu cộng, ta được:

\(A=\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{2014.2015}\)

\(A=2.\left(\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{2014.2015}\right)\)

\(A=2.\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-...+\frac{1}{2014}-\frac{1}{2015}\right)\)

\(A=2.\left(\frac{1}{3}-\frac{1}{2015}\right)\)

\(A=2.\frac{2012}{6045}\)

\(A=\frac{4024}{6045}\)

\(\frac{1}{2}A=\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2014.2015}=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2015}\)

=1/2-1/2015=2013/4030

=>A=2013/2015

tick nhé

a) ĐKXĐ : \(x\ne0\)

\(\left(-3+\frac{3}{x}-\frac{1}{3}\right):\left(1+\frac{2}{5}+\frac{2}{3}\right)=\frac{-5}{4}\)

\(\left(\frac{-9x}{3x}+\frac{9}{3x}-\frac{x}{3x}\right):\left(\frac{15}{15}+\frac{6}{15}+\frac{10}{15}\right)=\frac{-5}{4}\)

\(\frac{-9x+9-x}{3x}:\frac{15+6+10}{15}=\frac{-5}{4}\)

\(\frac{-10x+9}{3x}:\frac{31}{15}=\frac{-5}{4}\)

\(\frac{-10x+9}{3x}=\frac{-31}{12}\)

\(\Leftrightarrow12\left(-10x+9\right)=-31\cdot3x\)

\(\Leftrightarrow-120x+108=-93x\)

\(\Leftrightarrow-120x+93x=-108\)

\(\Leftrightarrow-27x=-108\)

\(\Leftrightarrow x=4\)

b) ĐKXĐ : \(x\ne0\)

\(\frac{-3x}{4}\cdot\left(\frac{1}{x}+\frac{2}{7}\right)=0\)

\(\Rightarrow\orbr{\begin{cases}\frac{-3x}{4}=0\\\frac{1}{x}+\frac{2}{7}=0\end{cases}}\)

\(\Rightarrow\orbr{\begin{cases}x=0\left(loai\right)\\\frac{-2}{-2x}=\frac{-2}{7}\end{cases}}\)

\(\Rightarrow\orbr{\begin{cases}x=0\left(loai\right)\\x=\frac{-7}{2}\end{cases}}\)

Vậy.....

c) phân tích ra rồi làm thôi e :)) a bận rồi