\(a_n=\frac{1-\frac{1}{6}\left(\frac{1-n}{1+n}\right)^{n-3}}{1+\frac{1}{6}\left(\frac{1-n}{1+n}\right)^{n-3}}\)

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

CMR với n thuộc N; n>=2 ta có:

\(A=\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right)...\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3}\)\(\frac{1}{3}\)

với \(a_1,a_2,a_3,.....,a_n>0;a_1+a_2+a_3+....+a_n=k\)

Chứng minh\(\left(a_1+\frac{1}{a_2}\right)^2+\left(a_2+\frac{1}{a_3}\right)^2+...+\left(a_n+\frac{1}{a_1}\right)^2\ge\frac{1}{n}\left(\frac{k^2+n^2}{k}\right)^2\)

Chứng minh:\(\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right)....\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3}\)

n thuộc n sao

chứng minh:

\(\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right).....\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3\left(n\right)thuộcNsao}\)

Cho a,b,c dương . CMR :
1) \(\frac{x^3}{y+z}+\frac{y^3}{x+z}+\frac{z^3}{x+y}\ge6;x+y+z\ge6\)
2) \(a_1.a_2....a_n\le\frac{1}{\left(n-1\right)^n};\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_n+1}=n-1\)
3) \(\frac{a}{b+c+1}+\frac{b}{a+c+1}+\frac{c}{b+a+1}+\left(1-a\right)\left(1-b\right)\left(1-c\right)\le1\) với a, b, c thuộc \(\left[0;1\right]\)