\(S=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}-\sqrt{1}}+\frac{1}{\sqrt{3}-\sqrt{2}}+...+\frac{1}{\sqrt{2025}-\sqrt{2024}}\)

Ta nhận xét thấy mỗi số hạng trong S đều dương. Từ đó ta đặt

\(A=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}-\sqrt{1}}+\frac{1}{\sqrt{3}-\sqrt{2}}+...+\frac{1}{\sqrt{2024}-\sqrt{2023}}\left(A>0\right)\)

\(\Rightarrow S=A+\frac{1}{\sqrt{2025}-\sqrt{2024}}=A+\frac{\sqrt{2025}+\sqrt{2024}}{\left(\sqrt{2025}-\sqrt{2024}\right)\left(\sqrt{2025}+\sqrt{2024}\right)}\)

\(=A+\sqrt{2025}+\sqrt{2024}>\sqrt{2025}=45\)

Vậy \(S>45\)

PS: Phan Thanh Tịnh xem lại bài giải nhé bạn

Ta có : 1 = (n + 1) - n =\(\left(\sqrt{n+1}\right)^2-\left(\sqrt{n}\right)^2\)

\(=\left(\sqrt{n+1}\right)^2-\sqrt{n+1}.\sqrt{n}+\sqrt{n+1}.\sqrt{n}+\left(\sqrt{n}\right)^2\)

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

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

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

Áp dụng vào bài toán,ta có :

\(S=\sqrt{1}+\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{2025}-\sqrt{2024}=\sqrt{2025}\)= 45

Vậy S = 45

vgfdgfd

Ko biết

\(\dfrac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\dfrac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{[\left(n+1\right)\sqrt{n}-n\sqrt{n+1}].[\left(n+1\right)\sqrt{n}+n\sqrt{n+1}]}\)

=\(\dfrac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)^2-n^2\left(n+1\right)}=\dfrac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)}=\dfrac{\sqrt{n}}{n}-\dfrac{\sqrt{n+1}}{n+1}\)

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

Áp dụng ta có S=\(\dfrac{1}{\sqrt{1}}-\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}-...+\dfrac{1}{\sqrt{2024}}-\dfrac{1}{\sqrt{2025}}=1-\dfrac{1}{\sqrt{2025}}=1-\dfrac{1}{45}=\dfrac{44}{45}\)

Ta có công thức tổng quát:

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

Vậy \(\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{4}}+...+\dfrac{1}{2025\sqrt{2024}+2024\sqrt{2025}}=\dfrac{1}{\sqrt{1}}-\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{3}}-\dfrac{1}{\sqrt{4}}+...+\dfrac{1}{\sqrt{2024}}-\dfrac{1}{\sqrt{2025}}=\dfrac{1}{\sqrt{1}}-\dfrac{1}{\sqrt{2025}}=1-\dfrac{1}{45}=\dfrac{44}{45}\)

1) Ta thấy:

\(4=1+3=1+\sqrt{9}\)

\(1+2\sqrt{2}=1+\sqrt{2^2\cdot2}=1+\sqrt{8}\)

Mà: \(\sqrt{8}< \sqrt{9}\)

\(\Rightarrow1+\sqrt{8}< 1+\sqrt{9}\)

\(\Rightarrow\dfrac{1}{1+\sqrt{8}}>\dfrac{1}{1+\sqrt{9}}\)

\(\Rightarrow\dfrac{1}{1+2\sqrt{2}}>\dfrac{1}{4}\)

2) Ta thấy:

\(2018< 2024\)

\(\Rightarrow\sqrt{2018}< \sqrt{2024}\) (1)

\(2025< 2026\)

\(\Rightarrow\sqrt{2025}< \sqrt{2026}\) (2)

Từ (1) và (2) ta có:

\(\sqrt{2018}+\sqrt{2025}< \sqrt{2024}+\sqrt{2026}\)

Xét biểu thức : \(\frac{1}{\sqrt{n}}=\frac{2}{\sqrt{n}+\sqrt{n}}>\frac{2}{\sqrt{n}+\sqrt{n+1}}=\frac{2\left(\sqrt{n+1}-\sqrt{n}\right)}{\left(\sqrt{n+1}\right)^2-\left(\sqrt{n}\right)^2}=2\left(\sqrt{n+1}-\sqrt{n}\right)\)với n > 0

Áp dụng : \(\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2024}}>2\left(\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{2025}-\sqrt{2024}\right)\)

\(\Rightarrow\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2024}}>2\left(\sqrt{2025}-1\right)=88\) (đpcm)

Bài 2: 

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

Tương tự ta có: \(\frac{1}{\sqrt{2}+\sqrt{3}}=\sqrt{3}-\sqrt{2}\);

\(\frac{1}{\sqrt{3}+\sqrt{4}}=\sqrt{4}-\sqrt{3}\); ............. ; \(\frac{1}{\sqrt{2024}+\sqrt{2025}}=\sqrt{2025}-\sqrt{2024}\)

\(\Rightarrow A=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+......+\sqrt{2025}-\sqrt{2024}\)

\(=\sqrt{2025}-\sqrt{1}=45-1=44\)

Bài 4: 

\(M=\frac{\sqrt{3-2\sqrt{2}}}{\sqrt{17-12\sqrt{2}}}-\frac{\sqrt{3+2\sqrt{2}}}{\sqrt{17+12\sqrt{2}}}\)

\(=\frac{\sqrt{2-2\sqrt{2}+1}}{\sqrt{9-2.3.2\sqrt{2}+8}}-\frac{\sqrt{2+2\sqrt{2}+1}}{\sqrt{9+2.3.2\sqrt{2}+8}}\)

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

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

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

\(=\left(3\sqrt{2}+\sqrt{16}-3-\sqrt{8}\right)-\left(3\sqrt{2}-\sqrt{16}+3-\sqrt{8}\right)\)

\(=3\sqrt{2}+4-3-\sqrt{8}-3\sqrt{2}+4-3+\sqrt{8}\)

\(=8-6=2\)là số tự nhiên

mình chỉ giải được phần này thôi

b.A = \(\sqrt{17}\)+\(\sqrt{26}\)+ 1 > \(\sqrt{16}\)+\(\sqrt{25}\)+ 1 = 4 + 5 +1 = 10

B = \(\sqrt{99}\)<\(\sqrt{100}\)= 10

=> A > B