giúp tui voi \(\sqrt{1+2013^2+\frac{2013^2}{2014^2}}+\frac{2013}{2014}\)
Những câu hỏi liên quan
Tính gía trị biểu thức:
\(A=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+....+\frac{1}{2014\sqrt{2013}+2013\sqrt{2014}}+\frac{1}{2015\sqrt{2014}+2014\sqrt{2015}}\)
Chứng minh \(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\) rồi áp dụng với n = 1,2,....,2014
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Tính giá trị của P = \(\sqrt{1+2013^2+\frac{2013^2}{2014^2}}\)+\(\frac{2013}{2014}\)
frac{frac{1}{2}+frac{1}{3}+......+frac{1}{2013}}{frac{2012}{1}+frac{2012}{2}+frac{2011}{3}+...+frac{1}{2013}}frac{frac{1}{2}+frac{1}{3}+..+frac{1}{2013}}{frac{2012}{1}+2+frac{2012}{2}+1+frac{2011}{3}+1+...+frac{1}{2013}+1-2014}frac{frac{1}{2}+frac{1}{3}+...+frac{1}{2013}}{frac{2014}{1}+frac{2014}{2}+...+frac{2014}{2013}-2014}frac{frac{1}{2}+frac{1}{3}+...+frac{1}{2013}}{2014left(1+frac{1}{2}+frac{1}{3}+...+frac{1}{2013}-1right)}frac{1}{2014}
Đọc tiếp
\(\frac{\frac{1}{2}+\frac{1}{3}+......+\frac{1}{2013}}{\frac{2012}{1}+\frac{2012}{2}+\frac{2011}{3}+...+\frac{1}{2013}}\)
=\(\frac{\frac{1}{2}+\frac{1}{3}+..+\frac{1}{2013}}{\frac{2012}{1}+2+\frac{2012}{2}+1+\frac{2011}{3}+1+...+\frac{1}{2013}+1-2014}\)
=\(\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}}{\frac{2014}{1}+\frac{2014}{2}+...+\frac{2014}{2013}-2014}\)
=\(\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}}{2014\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}-1\right)}\)
=\(\frac{1}{2014}\)
Giải pt
\(\sqrt{2x+\frac{2013-1}{\sqrt{2-x^2}}}-\sqrt[3]{2014-\frac{2013-1}{\sqrt{2-x^2}}}=\sqrt{x+2013}-\sqrt[3]{x+1}\)
Chứng minh số sau đây là số nguyên :
\(\sqrt{1^2+2013^2+\frac{2013^2}{2014^2}}\) +\(\frac{2013}{2014}\)
* Cách 1:
\(\sqrt{1^2+2013^2+\frac{2013^2}{2014^2}}\)
\(=\sqrt{2013^2.\left(1+\frac{1}{2013^2}+\frac{1}{2014^2}\right)}\)
\(=2013.\left(1+\frac{1}{2013}-\frac{1}{2014}\right)\)
\(=2013+1-\frac{2013}{2014}\)
\(=2014-\frac{2013}{2014}\)
* Cách 2:
\(\sqrt{1^2+2013^2+\frac{2013^2}{2014^2}}\)
\(=\sqrt{\left(1+2013\right)^2-2.2013+\frac{2013^2}{2014^2}}\)
\(=\sqrt{2014^2-2.2013+\left(\frac{2013}{2014}\right)^2}\)
\(=\sqrt{\left(2014-\frac{2013}{2014}\right)^2}\)
\(=2014-\frac{2013}{2014}\)
Từ 2 cách trên ta suy ra:
\(\sqrt{1^2+2013^2+\frac{2013^2}{2014^2}}+\frac{2013}{2014}\)
\(=2014-\frac{2013}{2014}+\frac{2013}{2014}\)
\(=2014\)
Theo đề bài trên, ta có thể suy ra công thức tổng quát như sau:
\(\sqrt{1^2+x^2+\frac{x^2}{\left(x+1\right)^2}}+\frac{x}{x+1}\)
(Chúc bạn học tốt và nhớ k cho mình với nhá!)
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Chứng mnh rằng \(\frac{2014}{\sqrt{2013}}+\frac{2013}{\sqrt{2014}}>\sqrt{2013}+\sqrt{2014}\)
Ta cần chứng minh:
\(\frac{2014}{\sqrt{2013}}+\frac{2013}{\sqrt{2014}}>\sqrt{2013}+\sqrt{2014}\)
\(\Leftrightarrow\frac{\sqrt{2013^3}+\sqrt{2014^3}}{\sqrt{2013}.\sqrt{2014}}>\sqrt{2013}+\sqrt{2014}\)
\(\Leftrightarrow\frac{\left(\sqrt{2013}+\sqrt{2014}\right)\left(2013-\sqrt{2013}.\sqrt{2014}+2014\right)}{\sqrt{2013}.\sqrt{2014}}>\sqrt{2013}+\sqrt{2014}\)
\(\Leftrightarrow\frac{2013-\sqrt{2013}.\sqrt{2014}+2014}{\sqrt{2013}.\sqrt{2014}}>1\)
\(\Leftrightarrow2013-2\sqrt{2013}.\sqrt{2014}+2014>0\)
\(\Leftrightarrow\left(\sqrt{2013}-\sqrt{2014}\right)^2>0\)đúng
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Tính giá trị biểu thức \(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2013}+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{2}{2013}+\frac{1}{2014}}\) .
\(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{1}{2014}}\)
Xét mẫu:
\(\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{1}{2014}\)
= \(\left(1+\frac{2013}{2}\right)+\left(1+\frac{2012}{3}\right)+...+\left(1+\frac{1}{2014}\right)+1\)
= \(\frac{2014}{2}+\frac{2014}{3}+....+\frac{2014}{2013}+\frac{2014}{2014}\)
= \(2014\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}\right)\)
\(\Rightarrow S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{2014.\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}\right)}\)
\(\Rightarrow S=\frac{1}{2014}\)
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Tính giá trị biểu thức \(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2013}+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{2}{2013}+\frac{1}{2014}}\) .
Tính giá trị biểu thức \(S=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2013}+\frac{1}{2014}}{\frac{2014}{1}+\frac{2013}{2}+\frac{2012}{3}+...+\frac{2}{2013}+\frac{1}{2014}}\) .