Question

Giúp đi giúp nha không khó đâu cũng không dễ đâu:

a) \(\dfrac{1}{2}< A< 1\)

biết A= \(\dfrac{1}{21}+\dfrac{1}{22}+...+\dfrac{1}{40}\)

b) \(\dfrac{1}{26}+\dfrac{1}{27}+\dfrac{1}{28}+...+\dfrac{1}{50}=1-\dfrac{1}{2}+...+\dfrac{1}{49}-\dfrac{1}{50}\)

Tick cho!

Answer 1

a. Ta có: \(\dfrac{1}{21}>\dfrac{1}{40};\dfrac{1}{22}>\dfrac{1}{40};...;\dfrac{1}{40}=\dfrac{1}{40}\)

\(\Rightarrow\dfrac{1}{21}+\dfrac{1}{22}+...+\dfrac{1}{40}>\dfrac{1}{40}+\dfrac{1}{40}+...+\dfrac{1}{40}\)(20 số hạng vì A có 20 số hạng)

\(\Rightarrow A>\dfrac{1}{40}.20\)

\(\Rightarrow A>\dfrac{1}{2}\left(1\right)\)

Ta lại có: \(\dfrac{1}{21}< \dfrac{1}{20};\dfrac{1}{22}< \dfrac{1}{20};...;\dfrac{1}{40}< \dfrac{1}{20}\)

\(\Rightarrow\dfrac{1}{21}+\dfrac{1}{22}+...+\dfrac{1}{40}< \dfrac{1}{20}+\dfrac{1}{20}+...+\dfrac{1}{20}\) (20 số hạng)

\(\Rightarrow A< \dfrac{1}{20}.20\)

\(\Rightarrow A< 1\left(2\right)\)

Từ \(\left(1\right)và\left(2\right)\) ta suy ra \(\dfrac{1}{2}< A< 1\)

Answer 2

b.Ta có: Đặt \(A=\dfrac{1}{26}+\dfrac{1}{27}+...+\dfrac{1}{50}\)

\(B=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{49}-\dfrac{1}{50}\)\(\Rightarrow B=\left(1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{49}\right)-\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{50}\right)\)

\(\Rightarrow B=1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{50}-2\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{50}\right)\)

\(\Rightarrow B=\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{50}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{25}\right)\)

\(\Rightarrow B=\dfrac{1}{25}+\dfrac{1}{26}+...+\dfrac{1}{50}=A\)

\(\Rightarrow B=A\left(đpcm\right)\)