Cho A=1/2^2+1/3^2 +1/4^2+…+1/150^2 CMR :A<1 13/10/2021 Bởi Sarah Cho A=1/2^2+1/3^2 +1/4^2+…+1/150^2 CMR :A<1
A = `1/$2^{2}$` + `1/$3^{2}$` + … + `1/$150^{2}$` Ta thấy : `1/$2^{2}$` < $\frac{1}{1.2}$ = 1 = `1/2` `1/$3^{2}$` < $\frac{1}{2.3}$ = `1/2` – `1/3` ………………. `1/$150^{2}$` < $\frac{1}{149.150}$ = `1/149` – `1/150` => A < $\frac{1}{1.2}$ + $\frac{1}{2.3}$ + … + $\frac{1}{149.150}$ => A < 1 – `1/2` + `1/2` – `1/3` + … + `1/149` – `1/150` => A < 1 – `1/150` Vì 1 – `1/150` < 1 => A < 1. Bình luận
Ta có: 2^2=2.2<1.2;3^2=3.3<2.3;4^2=4.4<3.4 suy ra A<1/2.3+1/3.4+…+1/149.150 =(1/2-1/3+1/3-1/4+…+1/149-1/150) =1/2-1/150 =75/150-1/150 =74/150 Vì 74/150<1 nên A<1 Vậy A<1 Bình luận
A = `1/$2^{2}$` + `1/$3^{2}$` + … + `1/$150^{2}$`
Ta thấy :
`1/$2^{2}$` < $\frac{1}{1.2}$ = 1 = `1/2`
`1/$3^{2}$` < $\frac{1}{2.3}$ = `1/2` – `1/3`
……………….
`1/$150^{2}$` < $\frac{1}{149.150}$ = `1/149` – `1/150`
=> A < $\frac{1}{1.2}$ + $\frac{1}{2.3}$ + … + $\frac{1}{149.150}$
=> A < 1 – `1/2` + `1/2` – `1/3` + … + `1/149` – `1/150`
=> A < 1 – `1/150`
Vì 1 – `1/150` < 1
=> A < 1.
Ta có: 2^2=2.2<1.2;3^2=3.3<2.3;4^2=4.4<3.4
suy ra A<1/2.3+1/3.4+…+1/149.150
=(1/2-1/3+1/3-1/4+…+1/149-1/150)
=1/2-1/150
=75/150-1/150
=74/150
Vì 74/150<1 nên A<1
Vậy A<1