Question

[Min]

Rút gọn các biểu thức (có tính quy luật)

A=\(\frac{1}{1\cdot\left(2n-1\right)}+\frac{1}{3\cdot\left(2n-3\right)}+....+\frac{1}{\left(2n-3\right)\cdot3}\)+\(\frac{1}{\left(2n-1\right)\cdot1}\)

giup mk nha

Answer 1

[Akai Haruma]

Lời giải:

\(A=\frac{1}{1(2n-1)}+\frac{1}{3(2n-3)}+...+\frac{1}{(2n-3).3}+\frac{1}{(2n-1).1}\)

\(2nA=\frac{1+(2n-1)}{1(2n-1)}+\frac{3+(2n-3)}{3(2n-3)}+....+\frac{(2n-3)+3}{(2n-3).3}+\frac{(2n-1)+1}{(2n-1).1}\)

\(2nA=\frac{1}{2n-1}+1+\frac{1}{2n-3}+\frac{1}{3}+...+\frac{1}{3}+\frac{1}{2n-3}+1+\frac{1}{2n-1}\)

\(=\left(\frac{1}{2n-1}+\frac{1}{2n-3}+...+\frac{1}{3}+1\right)+\left(1+\frac{1}{3}+...+\frac{1}{2n-3}+\frac{1}{2n-1}\right)\)

\(=2\left(1+\frac{1}{3}+...+\frac{1}{2n-1}\right)\)

\(\Rightarrow A=\frac{1}{n}\left(1+\frac{1}{3}+...+\frac{1}{2n-1}\right)\)