CM: [TEX]\frac{1}{41} + \frac{1}{42} + \frac{1}{43} +...++ \frac{1}{79} + \frac{1}{80} > \frac{7}{12}[/TEX]
Ta có:
[TEX]\frac{7}{12}[/TEX] = [TEX]\frac{4}{12}[/TEX] + [TEX]\frac{3}{12}[/TEX]
= [TEX]\frac{1}{3}[/TEX]+ [TEX]\frac{1}{4}[/TEX]
= [TEX]\frac{20}{60}[/TEX] + [TEX]\frac{20}{80}[/TEX]
Lại có:
[TEX]\frac{1}{41} + \frac{1}{42} + \frac{1}{43} +...+ \frac{1}{79} + \frac{1}{80} [/TEX]
= [TEX](\frac{1}{41}+ \frac{1}{42} + \frac{1}{43} + ...+ \frac{1}{43}) + (\frac{1}{61} + \frac{1}{62} +...+ \frac{1}{79} + \frac{1}{80})[/TEX]
Vì [TEX]\frac{1}{41}> \frac{1}{42} > \frac{1}{43} > ...> \frac{1}{59} > \frac{1}{60}[/TEX]
\Rightarrow ([TEX]\frac{1}{41} + \frac{1}{42} + \frac{1}{43} +...+\frac{1}{60}) > \frac{1}{60} + ...+ \frac{1}{60}[/TEX]
= [TEX]\frac{20}{60}[/TEX]
Ta có tiếp:
[TEX]\frac{1}{61} > \frac{1}{62}> ... > \frac{1}{79} > \frac{1}{80}[/TEX]
\Rightarrow ([TEX]\frac{1}{61} +\frac{1}{62}+ ... + \frac{1}{79} +\frac{1}{80}) > \frac{1}{80} + ...+ \frac{1}{80}[/TEX]
= [TEX]\frac{20}{80}[/TEX]
\Rightarrow [TEX]\frac{1}{41} + \frac{1}{42} + \frac{1}{43} +...+ \frac{1}{79} +\frac{1}{80} > \frac{20}{60} + \frac{20}{80}[/TEX] = [TEX]\frac{7}{12}[/TEX]
\Rightarrow [TEX]\frac{1}{41} + \frac{1}{42} + \frac{1}{43} +...+ \frac{1}{79} + \frac{1}{80} > \frac{7}{12}[/TEX] (đpcm)
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
