A = \(\frac{2012-1}{\sqrt{2012}}+\frac{2011+1}{\sqrt{2011}}=\sqrt{2012}-\frac{1}{\sqrt{2012}}+\sqrt{2011}+\frac{1}{\sqrt{2011}}\)

A = \(\sqrt{2012}+\sqrt{2011}+\left(\frac{1}{\sqrt{2011}}-\frac{1}{\sqrt{2012}}\right)=B+\left(\frac{1}{\sqrt{2011}}-\frac{1}{\sqrt{2012}}\right)\)

Mà 2011 < 2012 nên \(\frac{1}{\sqrt{2011}}>\frac{1}{\sqrt{2012}}\Rightarrow\frac{1}{\sqrt{2011}}-\frac{1}{\sqrt{2012}}>0\)

=> A > B

Ta có: \(\frac{1}{n\sqrt{n+1}+\left(n+1\right)\sqrt{n}}=\frac{1}{\sqrt{n}.\sqrt{n+1}\left(\sqrt{n}+\sqrt{n+1}\right)}\)

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

\(=\frac{\sqrt{n+1}}{\sqrt{n}.\sqrt{n+1}}-\frac{\sqrt{n}}{\sqrt{n}.\sqrt{n+1}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)

Thay n = 1, 2, 3, ..., 2011 vào C ta có:

\(C=1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2011}}-\frac{1}{\sqrt{2012}}=1-\frac{1}{\sqrt{2012}}\)

Vậy \(C=1-\frac{1}{\sqrt{2012}}.\)

uk xie xie (cảm ơn ) bạn , nhưng mik giải ra lâu r

Áp dụng BĐT Cô - si ngược dấu :

\(\sqrt{x-2010}=\frac{1}{2}\sqrt{4\left(x-2010\right)}\le\frac{4+\left(x-2010\right)}{4}\)

\(\Rightarrow\sqrt{x-2010}-1\le\frac{4+\left(x-2010\right)}{4}-1=\frac{x-2010}{4}\)

\(\Rightarrow\frac{\sqrt{x-2010}-1}{x-2010}\le\frac{1}{4}\)

Hoàn toàn tương tự với những phân thức còn lại 

\(\Rightarrow\frac{\sqrt{x-2010}-1}{x-2010}+\frac{\sqrt{y-2011}-1}{y-2011}\le\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}x-2010=4\\x-2011=4\\z-2012=4\end{cases}\Leftrightarrow\hept{\begin{cases}x=2014\\y=2015\\z=2016\end{cases}}}\)

Giải:

(1+1/2!)+(1+2/3!)+(1+3/4!)+....+(1+2011/2012!)=2011+(1/2!+2/3!+3/4!+...+2011/2012!)

=2011+(\(\frac{1}{2!}\)+\(\frac{3-1}{3!}\)+\(\frac{4-1}{4!}\)+...+\(\frac{2012-1}{2012!}\))= 2011 +(\(\frac{1}{2!}\)+\(\frac{1}{2!}\)-\(\frac{1}{3!}\)+\(\frac{1}{3!}\)-\(\frac{1}{4!}\)+...+\(\frac{1}{2011!}\)-\(\frac{1}{2012!}\))

= 2011+(1-\(\frac{1}{2012!}\))=2012 - \(\frac{1}{2012!}\)<2012 (đpcm)

cm nha

đăng ký tranghttps://www.youtube.com/channel/UCdMJRiuo_35tKETQtnAYOBQ để xem thêm nhiều bài tập nha

Xét biểu thức phụ : \(\frac{1}{\left(k+1\right)\sqrt{k}+k\left(\sqrt{k+1}\right)}=\frac{1}{\sqrt{k\left(k+1\right)}\left(\sqrt{k}+\sqrt{k+1}\right)}=\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k\left(k+1\right)}\left(k+1-k\right)}=\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k\left(k+1\right)}}=\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\)

Áp dụng : \(\frac{1}{2.\sqrt{1}+1.\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+\frac{1}{5\sqrt{4}+4\sqrt{5}}+...+\frac{1}{2012\sqrt{2011}+2011\sqrt{2012}}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2011}}-\frac{1}{\sqrt{2012}}=1-\frac{1}{\sqrt{2012}}\)

chóng váng

Dễ k cho mình trước rồi mình làm cho

K phai lop 7 nen k phai lam. Biet dau ma lam 

Mi nói bài toán hay nói tau

\(A=\frac{2!+\sqrt{3}}{2!}+\frac{3!+\sqrt{4}}{3!}+\frac{4!+\sqrt{5}}{4!}+....+\frac{2012!+\sqrt{2013}}{2012!}\)

\(=\frac{2!}{2!}+\frac{\sqrt{3}}{2!}+\frac{3!}{3!}+\frac{\sqrt{4}}{3!}+.....+\frac{2012!}{2012!}+\frac{\sqrt{2013}}{2012!}\)

\(=2012+\left(\frac{\sqrt{3}}{2!}+\frac{\sqrt{4}}{3!}+....+\frac{\sqrt{2011}}{2012!}\right)\)

Mà \(\frac{\sqrt{3}}{2!}+\frac{\sqrt{4}}{3!}+...+\frac{\sqrt{2013}}{2012!}>0\)

\(\Rightarrow A>2012+0=2012\)

Đề sai nên t sửa lại r nhé

haizzzz đệ lm sai rồi kìa =((

đệ sai chỗ nào ah sư phụ?