Đk: x \(\ge\)0 và x \(\ne\)1

a)\(P=\frac{3\left(x+\sqrt{x}-3\right)}{x+\sqrt{x}-2}+\frac{\sqrt{x}+3}{\sqrt{x}+2}-\frac{\sqrt{x}-2}{\sqrt{x}-1}\)

\(P=\frac{3x+3\sqrt{x}-9+\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)

\(P=\frac{3x+3\sqrt{x}-9+x+2\sqrt{x}-3-x+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)

\(P=\frac{3x+5\sqrt{x}-8}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\frac{3x+8\sqrt{x}-3\sqrt{x}-8}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)

\(P=\frac{\left(3\sqrt{x}+8\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}=\frac{3\sqrt{x}+8}{\sqrt{x}+2}\)

b) Với x \(\ge\)0 và x \(\ne\)1 (1), ta có:

\(P< \frac{15}{4}\) <=> \(\frac{3\sqrt{x}+8}{\sqrt{x}+2}< \frac{15}{4}\) <=> \(12\sqrt{x}+32< 15\sqrt{x}+30\)

<=> \(3\sqrt{x}>2\) <=> \(\sqrt{x}>\frac{2}{3}\) <=> \(x>\frac{4}{9}\)(2)

Từ (1) và (2) => \(x>\frac{4}{9}\) và x \(\ne\)1

\(A=\frac{\sqrt{x}}{\sqrt{x}+3}+\frac{2\sqrt{x}}{\sqrt{x}-3}-\frac{3x+9}{x-9}\)

\(A=\frac{\sqrt{x}\left(\sqrt{x}-3\right)+2\sqrt{x}\left(\sqrt{x}+3\right)-3x-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(A=\frac{x-3\sqrt{x}+2x+6\sqrt{x}-3x-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)

\(A=\frac{3\sqrt{x}-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}=\frac{3\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}=\frac{3}{\sqrt{x}+3}\)

b) Với x \(\ge\)0; x khác 9

Ta có: \(A=\frac{1}{3}\) <=> \(\frac{3}{\sqrt{x}+3}=\frac{1}{3}\) <=> \(\sqrt{x}+3=9\) <=> \(\sqrt{x}=6\) <=> x=  36 (tm)

c) Ta có: \(\sqrt{x}+3\ge3\) => \(\frac{3}{\sqrt{x}+3}\le\frac{3}{3}=1\) => A \(\le\)1

Dấu "=" xảy ra<=> x = 0

Vậy MaxA = 1 <=> x = 0

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

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

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

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

\(T=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{100\sqrt{99}+99\sqrt{100}}\)

\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{99}}-\frac{1}{\sqrt{100}}\)

\(=1-\frac{1}{\sqrt{100}}=\frac{9}{10}\)

b) \(\Leftrightarrow1-\frac{1}{\sqrt{n+1}}=\frac{4}{5}\)( làm tắt nhé )

\(\Leftrightarrow\frac{1}{\sqrt{n+1}}=\frac{1}{5}\)

\(\Leftrightarrow\sqrt{n+1}=5\)

\(\Leftrightarrow n=24\)

Vậy ...

\(A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}\)

\(2A=\frac{2}{\sqrt{1}+\sqrt{2}}+\frac{2}{\sqrt{3}+\sqrt{4}}+\frac{2}{\sqrt{5}+\sqrt{6}}+...+\frac{2}{\sqrt{79}+\sqrt{80}}\)

\(>\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}+\frac{1}{\sqrt{80}+\sqrt{81}}\)

\(=\frac{\sqrt{2}-1}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+\frac{\sqrt{4}-\sqrt{3}}{4-3}+...+\frac{\sqrt{81}-\sqrt{80}}{81-80}\)

\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+...+\sqrt{81}-\sqrt{80}\)

\(=\sqrt{81}-1=8\)

suy ra đpcm. 

\(A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{80}+\sqrt{81}}\)

\(=\frac{\sqrt{2}-1}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+\frac{\sqrt{4}-\sqrt{3}}{4-3}+...+\frac{\sqrt{81}-\sqrt{80}}{81-80}\)

\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+...+\sqrt{81}-\sqrt{80}\)

\(=\sqrt{81}-1=8\)

\(B=\frac{1}{\sqrt{1}-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}-...-\frac{1}{\sqrt{24}-\sqrt{25}}\)

\(=\frac{\sqrt{1}+\sqrt{2}}{1-2}-\frac{\sqrt{2}+\sqrt{3}}{2-3}+\frac{\sqrt{3}+\sqrt{4}}{3-4}-...-\frac{\sqrt{24}+\sqrt{25}}{24-25}\)

\(=-\sqrt{1}-\sqrt{2}+\sqrt{2}+\sqrt{3}-\sqrt{3}-\sqrt{4}+...+\sqrt{25}+\sqrt{24}\)

\(=\sqrt{25}-\sqrt{1}=4\)

bằng 8

nhanh nha mn