Menu Close

prove-that-1-2-3-4-5-6-7-8-9-10-99-100-lt-1-10-




Question Number 193785 by aba last updated on 19/Jun/23
prove that  (1/2)×(3/4)×(5/6)×(7/8)×(9/(10))×...×((99)/(100))<(1/(10))
$${prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{5}}{\mathrm{6}}×\frac{\mathrm{7}}{\mathrm{8}}×\frac{\mathrm{9}}{\mathrm{10}}×…×\frac{\mathrm{99}}{\mathrm{100}}<\frac{\mathrm{1}}{\mathrm{10}}\:\:\: \\ $$
Answered by MM42 last updated on 20/Jun/23
a=(1/2)×(3/4)×(5/6)...×((99)/(100))<  <(2/3)×(4/5)×(6/7)×...×((98)/(99))×((100)/(101))=  =(1/((3/2)×(5/4)×(7/6)×...×((99)/(98))×((101)/(100))))  ⇒a<(1/(101a))⇒a<(1/( (√(101))))<(1/(10)) ✓
$${a}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{5}}{\mathrm{6}}…×\frac{\mathrm{99}}{\mathrm{100}}< \\ $$$$<\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{4}}{\mathrm{5}}×\frac{\mathrm{6}}{\mathrm{7}}×…×\frac{\mathrm{98}}{\mathrm{99}}×\frac{\mathrm{100}}{\mathrm{101}}= \\ $$$$=\frac{\mathrm{1}}{\frac{\mathrm{3}}{\mathrm{2}}×\frac{\mathrm{5}}{\mathrm{4}}×\frac{\mathrm{7}}{\mathrm{6}}×…×\frac{\mathrm{99}}{\mathrm{98}}×\frac{\mathrm{101}}{\mathrm{100}}} \\ $$$$\Rightarrow{a}<\frac{\mathrm{1}}{\mathrm{101}{a}}\Rightarrow{a}<\frac{\mathrm{1}}{\:\sqrt{\mathrm{101}}}<\frac{\mathrm{1}}{\mathrm{10}}\:\checkmark \\ $$
Commented by aba last updated on 20/Jun/23
perfect
$$\mathrm{perfect} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *