Question

[Hoàng Phúc]

So sánh A và B(ko dùng máy tính)

A=\(\sqrt{481}\)

\(B=\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+..+\frac{1}{\sqrt{2013}+\sqrt{2015}}\)

Minh Triều làm giúp đi,tick cho

Answer 1

[Đặng Minh Triều]

Ta thấy: \(\left(\sqrt{a}+\sqrt{b}\right)^2=a+b+2\sqrt{ab}\)

\(\left(\sqrt{a+b}\right)^2=a+b\)

Nếu: \(2\sqrt{ab}>0\left(a,b>0\right)\text{ thì: }\left(\sqrt{a}+\sqrt{b}\right)^2>\left(\sqrt{a+b}\right)^2\)

<=>\(\sqrt{a}+\sqrt{b}>\sqrt{a+b}\)

\(B=\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{5}}+....+\frac{1}{\sqrt{2013}+\sqrt{2015}}\)

\(=\frac{1}{2}.\left(\frac{2}{\sqrt{1}+\sqrt{3}}+\frac{2}{\sqrt{3}+\sqrt{5}}+...+\frac{2}{\sqrt{2013}-\sqrt{2014}}\right)\)

\(=\frac{1}{2}.\left(-1+\sqrt{3}-\sqrt{3}+\sqrt{5}-...-\sqrt{2013}+\sqrt{2015}\right)\)

=\(\frac{\sqrt{2015}-1}{2}\)

Xét hiệu: B-A=\(\frac{\sqrt{2015}-1}{2}-\sqrt{481}=\frac{\sqrt{2015}-1}{2}-\frac{\sqrt{1924}}{2}=\frac{\sqrt{2015}-\left(\sqrt{1}+\sqrt{1924}\right)}{2}>\frac{\sqrt{2015}-\sqrt{1+1924}}{2}\)

\(=\frac{\sqrt{2015}-\sqrt{1925}}{2}>0\Rightarrow A>B\)

Answer 2

[Đặng Minh Triều]

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