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

p r o v e t h a t

\frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times \frac{7}{8} \times \frac{9}{10} \times \dots \times \frac{99}{100} < \frac{1}{10}

Answered by MM42 last updated on 20/Jun/23

a = \frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \dots \times \frac{99}{100} <

< \frac{2}{3} \times \frac{4}{5} \times \frac{6}{7} \times \dots \times \frac{98}{99} \times \frac{100}{101} =

= \frac{1}{\frac{3}{2} \times \frac{5}{4} \times \frac{7}{6} \times \dots \times \frac{99}{98} \times \frac{101}{100}}

\Rightarrow a < \frac{1}{101 a} \Rightarrow a < \frac{1}{\sqrt{101}} < \frac{1}{10} ✓

Commented by aba last updated on 20/Jun/23

perfect

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