Đặt \(\left\{{}\begin{matrix}\sqrt{x}=a\\\sqrt{y}=b\end{matrix}\right.\), ta có:

\(A=\left[\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\times\dfrac{2}{a+b}+\dfrac{1}{a^2}+\dfrac{1}{b^2}\right]\)\(\times\dfrac{a^3+ab^2+a^2b+b^3}{ab^3+a^3b}\)

\(=\left(\dfrac{b+a}{ab}\times\dfrac{2}{a+b}+\dfrac{b^2+a^2}{a^2b^2}\right)\)\(\times\dfrac{a^2\left(a+b\right)+b^2\left(a+b\right)}{ab\left(a^2+b^2\right)}\)

\(=\dfrac{2ab+b^2+a^2}{a^2b^2}\times\dfrac{\left(a+b\right)\left(a^2+b^2\right)}{ab\left(b^2+a^2\right)}\)

\(=\dfrac{\left(a+b\right)^3}{a^3b^3}\)

\(=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^3}{\sqrt{\left(xy\right)^3}}\)

Đặt \(A=\dfrac{n}{4+n^4}\)

\(=\dfrac{n}{n^4+4n^2+4-4n^2}\)

\(=\dfrac{n}{\left(n^2+2\right)^2-\left(2n\right)^2}\)

\(=\dfrac{n}{\left(n^2+2-2n\right)\left(n^2+2+2n\right)}\)

\(\Rightarrow4A=\dfrac{4n}{\left(n^2-2n+2\right)\left(n^2+2n+2\right)}\)

\(=\dfrac{1}{n^2-2n+2}-\dfrac{1}{n^2+2n+2}\)

Đặt \(P=\dfrac{1}{4+1^4}+\dfrac{3}{4+3^4}+...+\dfrac{2n-1}{4+\left(2n-1\right)^4}\)

\(\Rightarrow4P=\dfrac{4}{4+1^4}+\dfrac{12}{4+3^4}+...+\dfrac{4\left(2n-1\right)}{4+\left(2n-1\right)^4}\)

\(=\dfrac{1}{1^2-2\times1+2}-\dfrac{1}{1^2+2\times1+2}\)

\(+\dfrac{1}{3^2-2\times3+2}-\dfrac{1}{3^2+2\times3+2}+...+\)

\(\dfrac{1}{\left(2n-1\right)^2-2\left(2n-1\right)+2}-\dfrac{1}{\left(2n-1\right)^2+2\left(2n-1\right)+2}\)

\(=1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{17}+...+\)

\(\dfrac{1}{\left(2n-1\right)^2-2\left(2n-1\right)+2}-\dfrac{1}{4n^2-4n+1+4n-2+2}\)

\(=1-\dfrac{1}{4n^2+1}\)

\(\Rightarrow P=\dfrac{1}{4}-\dfrac{1}{4\left(4n^2+1\right)}\)

Tên: Thiên Băng

Lớp: 8

Link: Góc học tập của Thiên Băng | Học trực tuyến

Áp dụng bđt Cauchy, ta có:

\(\dfrac{x^2}{y^2}+\dfrac{y^2}{z^2}+\dfrac{z^2}{x^2}\ge\sqrt{\dfrac{x^2}{y^2}\times\dfrac{y^2}{z^2}}+\sqrt{\dfrac{y^2}{z^2}\times\dfrac{z^2}{x^2}}+\sqrt{\dfrac{x^2}{y^2}\times\dfrac{z^2}{x^2}}=\dfrac{x}{z}+\dfrac{y}{x}+\dfrac{z}{y}\left(\text{đ}pcm\right)\)

Dấu "=" xảy ra khi x = y = z

Mark the letter A, B, C, or D on your answer sheet to indicate the word that differs from the other three in the position of primary stress in each of the following questions.

A. ability

B. sympathetic

C. political

D. significant