Question

Bài 2 Cho M=1/31+1/32+...+1/89+1/90.So sanh M với 5/6

Giup vs mk cần gấp mk like cho

Answer 1

\(M=\dfrac{1}{31}+\dfrac{1}{32}+.................+\dfrac{1}{89}+\dfrac{1}{90}\)

\(\Leftrightarrow M=\left(\dfrac{1}{31}+\dfrac{1}{32}+.........+\dfrac{1}{60}\right)+\left(\dfrac{1}{61}+\dfrac{1}{62}+........+\dfrac{1}{90}\right)\)

Đặt :

\(A=\dfrac{1}{31}+\dfrac{1}{32}+.......+\dfrac{1}{60}>\dfrac{1}{60}+\dfrac{1}{60}+.......+\dfrac{1}{60}=\dfrac{1}{60}.30=\dfrac{1}{2}\)

\(B=\dfrac{1}{61}+\dfrac{1}{62}+.......+\dfrac{1}{90}>\dfrac{1}{90}+\dfrac{1}{90}+......+\dfrac{1}{90}=\dfrac{1}{90}.30=\dfrac{1}{3}\)

\(\Leftrightarrow M=\dfrac{1}{31}+\dfrac{1}{32}+.........+\dfrac{1}{89}+\dfrac{1}{90}=A+B< \dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}\)

\(\Leftrightarrow M< \dfrac{5}{6}\rightarrowđpcm\)