Question

Đã trả lời bởi chuyên gia

Answer 1

HD:

$2 B = \frac{1}{1 + \sqrt{2}} + \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{5} + \sqrt{6}} + . . . + \frac{1}{\sqrt{79} + \sqrt{80}} + \frac{1}{\sqrt{79} + \sqrt{80}} t a c ó \sqrt{n} < \sqrt{n + 2} \Rightarrow \sqrt{n} + \sqrt{n + 1} < \sqrt{n + 1} + \sqrt{n + 2} \Rightarrow \frac{1}{\sqrt{n} + \sqrt{n + 1}} > \frac{1}{\sqrt{n + 1} + \sqrt{n + 2}} k h i đ ó : 2 B > \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{6} + \sqrt{7}} + . . . + \frac{1}{\sqrt{79} + \sqrt{80}} + \frac{1}{\sqrt{81} + \sqrt{80}} M ặ t k h á c : \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{6} + \sqrt{7}} + . . . + \frac{1}{\sqrt{79} + \sqrt{80}} + \frac{1}{\sqrt{81} + \sqrt{80}} = \frac{\sqrt{2} - 1}{2 - 1} + \frac{\sqrt{3} - \sqrt{2}}{3 - 2} + \frac{\sqrt{4} - \sqrt{3}}{4 - 3} + \frac{\sqrt{5} - \sqrt{4}}{5 - 4} + . . . . + \frac{\sqrt{80} - \sqrt{79}}{80 - 79} + \frac{\sqrt{81} - \sqrt{80}}{81 - 80} = \sqrt{2} - 1 + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} + \sqrt{5} - \sqrt{4} + . . . + \sqrt{80} - \sqrt{79} + \sqrt{81} - \sqrt{80} = \sqrt{81} - 1 = 9 - 1 = 8 s u y r a 2 B > 8 h a y B > 4 (đ p c m)$

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Answer 2

HD:

$2 B = \frac{1}{1 + \sqrt{2}} + \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{5} + \sqrt{6}} + . . . + \frac{1}{\sqrt{79} + \sqrt{80}} + \frac{1}{\sqrt{79} + \sqrt{80}} t a c ó \sqrt{n} < \sqrt{n + 2} \Rightarrow \sqrt{n} + \sqrt{n + 1} < \sqrt{n + 1} + \sqrt{n + 2} \Rightarrow \frac{1}{\sqrt{n} + \sqrt{n + 1}} > \frac{1}{\sqrt{n + 1} + \sqrt{n + 2}} k h i đ ó : 2 B > \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{6} + \sqrt{7}} + . . . + \frac{1}{\sqrt{79} + \sqrt{80}} + \frac{1}{\sqrt{81} + \sqrt{80}} M ặ t k h á c : \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{6} + \sqrt{7}} + . . . + \frac{1}{\sqrt{79} + \sqrt{80}} + \frac{1}{\sqrt{81} + \sqrt{80}} = \frac{\sqrt{2} - 1}{2 - 1} + \frac{\sqrt{3} - \sqrt{2}}{3 - 2} + \frac{\sqrt{4} - \sqrt{3}}{4 - 3} + \frac{\sqrt{5} - \sqrt{4}}{5 - 4} + . . . . + \frac{\sqrt{80} - \sqrt{79}}{80 - 79} + \frac{\sqrt{81} - \sqrt{80}}{81 - 80} = \sqrt{2} - 1 + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} + \sqrt{5} - \sqrt{4} + . . . + \sqrt{80} - \sqrt{79} + \sqrt{81} - \sqrt{80} = \sqrt{81} - 1 = 9 - 1 = 8 s u y r a 2 B > 8 h a y B > 4 (đ p c m)$

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