a, Cho A=1/11+1/12+1/13+...+1/70. CMR: 4/3<A<5/2
[tex]A=\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{70}[/tex]
[tex]A=(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20})+(\frac{1}{21}+\frac{1}{22}+...+\frac{1}{30})+(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{70})[/tex]
Có:
+)[tex]\frac{1}{10}+\frac{1}{10}+...+\frac{1}{10}>\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}>\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}[/tex]
[tex]\Rightarrow 1>\frac{1}{11}+\frac{1}{12}+...+\frac{1}{20}>\frac{1}{2}[/tex] (1)
+)[tex]\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}>\frac{1}{21}+\frac{1}{22}+...+\frac{1}{30}>\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}[/tex]
[tex]\Rightarrow \frac{1}{2}>\frac{1}{21}+\frac{1}{22}+...+\frac{1}{30}>\frac{1}{3}[/tex] (2)
+)[tex]\frac{1}{30}>\frac{1}{61}+\frac{1}{62}
\Rightarrow 10.(\frac{1}{30})>\frac{1}{61}+\frac{1}{62}+...+\frac{1}{70}[/tex]
[tex]20.(\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30})+10.\frac{1}{30}>\frac{1}{31}+\frac{1}{32}+...+\frac{1}{70}>\frac{1}{70}+\frac{1}{70}+...+\frac{1}{70}[/tex]
[tex]\Rightarrow 1>\frac{1}{31}+\frac{1}{32}+...+\frac{1}{70}>\frac{4}{7}[/tex] (3)
Từ (1), (2) và (3)[tex]\Rightarrow 1+\frac{1}{2}+1>\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{70}>\frac{4}{7}+\frac{1}{3}+\frac{1}{2}[/tex]
[tex]\Rightarrow \frac{5}{2}>A>\frac{59}{42}[/tex]
[tex]\Rightarrow \frac{5}{2}>A>\frac{4}{3}[/tex]
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
