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\(a_n=\frac{1-\frac{1}{6}\left(\frac{1-n}{1+n}\right)^{n-3}}{1+\frac{1}{6}\left(\frac{1-n}{1+n}\right)^{n-3}}\)
cho n số thực dương a_{_{ }1},a_2,...,a_ncó tổng bằng 1. Chứng minh rằng:a) 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)^2geleft(frac{n^2+1}{n}right)^2b) left(a_1+frac{1}{a_1}right)^2+left(a_2+frac{1}{a_2}right)^2+...+left(a_n+frac{1}{a_n}right)^2geleft(frac{n^2+1}{n}right)^2
Đọc tiếp
cho n số thực dương \(a_{_{ }1},a_2,...,a_n\)có tổng bằng 1. Chứng minh rằng:
a) \(\left(a_1+\frac{1}{a_2}\right)^2+\left(a_2+\frac{1}{a_3}\right)^2+...+\left(a_n+\frac{1}{a_1}\right)^2\ge\left(\frac{n^2+1}{n}\right)^2\)
b) \(\left(a_1+\frac{1}{a_1}\right)^2+\left(a_2+\frac{1}{a_2}\right)^2+...+\left(a_n+\frac{1}{a_n}\right)^2\ge\left(\frac{n^2+1}{n}\right)^2\)
\(a_n=\frac{1+\left(\frac{n}{n+2}\right)^n}{1-\left(\frac{n}{n+2}\right)^n}\)
CMR với n thuộc N; n>=2 ta có:
\(A=\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right)...\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3}\)\(\frac{1}{3}\)
chứng minh:
\(\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right).....\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3\left(n\right)thuộcNsao}\)
Chứng minh với mọi số nguyên dương n, n>=2 ta có :
\(\left(1-\frac{2}{6}\right)\left(1-\frac{2}{12}\right)\left(1-\frac{2}{20}\right)...\left(1-\frac{2}{n\left(n+1\right)}\right)>\frac{1}{3}\)
\(\sin^3\frac{x}{3}+3\sin^3\frac{x}{3^2}+...+3^{n-1}\sin^3\frac{x}{3}=\frac{1}{4}\left(3^n\sin^3\frac{x}{3^n}-\sin x\right)\)\(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{2n+1}{2n+2}<\frac{1}{\sqrt{3n+4}}\left(n\ge1\right)\)\(\left(n!\right)^2\ge n^2\ge\left(n+1\right)^{n-1}cho\left(n\ge1\right)\)
11 tháng 9 2017 lúc 19:40
Cho n số dương a1,a2 ,...,an. Chứng minh rằng :
\(\left(a_1+a_2+...+a_n\right)\left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)\ge n^2\)
20 tháng 6 2018 lúc 11:51
Cho \(\hept{\begin{cases}a_1>a_2>...>a_n>0\\1\le k\in Z\end{cases}}\)
CMR : \(a_1+\frac{1}{a_n\left(a_1-a_2\right)^k\left(a_2-a_3\right)^k...\left(a_{n-1}-a_n\right)^k}\ge\frac{\left(n-1\right)k+2}{\sqrt[\left(n-1\right)k+2]{k^{\left(n-1\right)k}}}\)