Cho A = 1/21 + 1/22 + 1/23 + … + 1/80 chứng minh 1 < A < 2
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Đặt $A=\dfrac1{21}+\dfrac1{22}+\dfrac1{23}+…+\dfrac1{80}$Ta có : $A=(\dfrac1{21}+\dfrac1{22}+…+\dfrac1{40})+(\dfrac1{41}+…+\dfrac1{80})$Ta thấy :1/21<1/201/22<1/20…1/40<1/201/41<1/401/42<1/40…1/80<1/40⇒$A>(\dfrac1{40}+\dfrac1{40}+…+\dfrac1{40})+(\dfrac1{80}+..+\dfrac1{80})$⇒$\ A>\dfrac{20}{40}+\dfrac{40}{80}$⇒$\ A>\dfrac12+\dfrac12$=2/2=1⇒$\ A>1 $ (1)Ta lại có : $A=(\dfrac1{21}+\dfrac1{22}+…+\dfrac1{40})+(\dfrac1{41}+…+\dfrac1{80})$⇒$\ A<(\dfrac1{20}+\dfrac1{20}+…+\dfrac1{20})+(\dfrac1{40}+…+\dfrac1{40})$⇒$\to A<\dfrac{20}{20}+\dfrac{40}{40}=$1+1=2⇒$\to A<2$ (2)Kết hợp (1) và (2) , ta được :1<A<2 (đpcm)
