Question

Tính giá trị biểu thức: \(A=\dfrac{1}{1\sqrt{3}+3\sqrt{1}}+\dfrac{1}{3\sqrt{5}+5\sqrt{3}}+\dfrac{1}{5\sqrt{7}+7\sqrt{5}}+...+\dfrac{1}{79\sqrt{81}+81\sqrt{79}}\)

Answer 1

Với mọi \(n\in\text{ℕ*}\), ta có:

\(\dfrac{2}{n\sqrt{n+2}+\left(n+2\right)\sqrt{n}}\)\(=\dfrac{2\left(n\sqrt{n+2}-\left(n+2\right)\sqrt{n}\right)}{\left(n+2\right)^2n-n^2\left(n+2\right)}\)\(=\dfrac{2\left[\left(n+2\right)\sqrt{n}-n\sqrt{n+2}\right]}{n\left(n+2\right)}\)\(=\dfrac{1}{\sqrt{n}}-\dfrac{1}{\sqrt{n+2}}\)

Vậy ta có:

\(2A=\dfrac{1}{\sqrt{1}}-\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{3}}-\dfrac{1}{\sqrt{5}}+...-\dfrac{1}{\sqrt{81}}\)

\(=1-\dfrac{1}{\sqrt{81}}\)

\(A=\dfrac{1-\dfrac{1}{\sqrt{81}}}{2}\)