Question

chứng tỏ rằng : 1/11+1/12+1/13+...+1/69+1/70>1+5/29

Ai làm đúng và nhanh nhất mình sẽ tick max điểm

Answer 1

Gọi \(\dfrac{1}{11}+\dfrac{1}{12}+\dfrac{1}{13}+...+\dfrac{1}{69}+\dfrac{1}{70}\) là \(S\)

Ta nhận thấy:

\(\dfrac{1}{11},\dfrac{1}{12},\dfrac{1}{13},...,\dfrac{1}{19}\)đều lớn hơn \(\dfrac{1}{20}\)

\(\dfrac{1}{21},\dfrac{1}{22},\dfrac{1}{23},...,\dfrac{1}{29}\)đều lớn hơn \(\dfrac{1}{30}\) \(\dfrac{1}{31},\dfrac{1}{32},\dfrac{1}{33},...,\dfrac{1}{39}\)đều lớn hơn \(\dfrac{1}{40}\) \(\dfrac{1}{41},\dfrac{1}{42},\dfrac{1}{43},...,\dfrac{1}{49}\)đều lớn hơn \(\dfrac{1}{50}\) \(\dfrac{1}{51},\dfrac{1}{52},\dfrac{1}{53},...,\dfrac{1}{59}\)đều lớn hơn \(\dfrac{1}{60}\)

\(\dfrac{1}{61},\dfrac{1}{62},\dfrac{1}{63},...,\dfrac{1}{69}\)đều lớn hơn \(\dfrac{1}{70}\)

\(\Rightarrow S< \dfrac{1}{20}+\dfrac{1}{20}+...+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{30}+...+\dfrac{1}{30}+\dfrac{1}{40}+\dfrac{1}{40}+...+\dfrac{1}{40}+\dfrac{1}{50}+\dfrac{1}{50}+...+\dfrac{1}{50}+\dfrac{1}{60}+\dfrac{1}{60}+...+\dfrac{1}{60}+\dfrac{1}{70}+\dfrac{1}{70}+...+\dfrac{1}{70}\\ \Leftrightarrow S< \dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}\\ =\dfrac{223}{140}\) \(1\dfrac{5}{29}=\dfrac{34}{29}\) \(\dfrac{223}{140}>\dfrac{210}{140}=\dfrac{3}{2}=\dfrac{87}{58}>\dfrac{34}{29}\) Vậy \(\dfrac{1}{11}+\dfrac{1}{12}+\dfrac{1}{13}+...+\dfrac{1}{69}+\dfrac{1}{70}>1+\dfrac{5}{29}\left(đpcm\right)\)