Cho A=1/2-1/3+1/4-1/5+…+1/48-1/49 Chứng tỏ 1/5 18/10/2021 Bởi Raelynn Cho A=1/2-1/3+1/4-1/5+…+1/48-1/49 Chứng tỏ 1/5 { "@context": "https://schema.org", "@type": "QAPage", "mainEntity": { "@type": "Question", "name": " Cho A=1/2-1/3+1/4-1/5+...+1/48-1/49 Chứng tỏ 1/5
`A = 1/2 – 1/3 + 1/4 – 1/5 + … + 1/48 – 1/49` `1 – A – 1/50 = 1 – 1/2 + 1/3 – 1/4 + 1/5 + … + 1/49 – 1/50` `49/50 – A = 1 + 1/2 + 1/3 + 1/4 + … + 1/49 + 1/50 – 2(1/2 + 1/4 + 1/6 + … + 1/50)` `49/50 – A = 1 + 1/2 + 1/3 + 1/4 + … + 1/49 + 1/50 – 1 – 1/2 – 1/3 – 1/4 – … – 1/25` `49/50 – A = 1/26 + 1/27 + 1/28 + … + 1/50` `A = 49/50 – (1/26 + 1/27 + 1/28 + … + 1/50)` $\text{Ta có : }$ `1/26 > 1/30` `1/27 > 1/30` `………………` `1/29 > 1/30` `1/31 > 1/40` `1/32 > 1/40` `…………….` `1/39 > 1/40` `1/41 > 1/50` `1/42 > 1/50` `……………..` `1/49 > 1/50` `=> 1/26 + 1/27 + … + 1/50 > 5 . 1/30 + 10 . 1/40 + 10 . 1/50 = 1/6 + 1/4 + 1/5 = 37/60 > 36/60 = 3/5` `=> 49/50 – (1/26 + 1/27 + … + 1/50) < 49/50 – 3/5 = 19/50 < 20/50 = 2/5` $\text{Ta có : }$ `1/26 < 1/25` `1/27 < 1/25` `……………` `1/30 < 1/25` `1/31 < 1/30` `1/32 < 1/30` `……………` `1/40 < 1/30` `1/41 < 1/40` `1/42 < 1/40` `…………..` `1/50 < 1/40` `=> 1/26 +1/27 + … + 1/50 < 5 . 1/25 + 10 . 1/30 + 10 . 1/40 = 47/40 < 48/60 = 4/5` `=> 49/50 – (1/26 + 1/27 + … + 1/50) > 49/50 – 4/5 = 9/50 > 10/50 = 1/5.` $\text{Vậy }$`1/5 < A < 2/5.` Bình luận
`A = 1/2 – 1/3 + 1/4 – 1/5 + … + 1/48 – 1/49`
`1 – A – 1/50 = 1 – 1/2 + 1/3 – 1/4 + 1/5 + … + 1/49 – 1/50`
`49/50 – A = 1 + 1/2 + 1/3 + 1/4 + … + 1/49 + 1/50 – 2(1/2 + 1/4 + 1/6 + … + 1/50)`
`49/50 – A = 1 + 1/2 + 1/3 + 1/4 + … + 1/49 + 1/50 – 1 – 1/2 – 1/3 – 1/4 – … – 1/25`
`49/50 – A = 1/26 + 1/27 + 1/28 + … + 1/50`
`A = 49/50 – (1/26 + 1/27 + 1/28 + … + 1/50)`
$\text{Ta có : }$
`1/26 > 1/30`
`1/27 > 1/30`
`………………`
`1/29 > 1/30`
`1/31 > 1/40`
`1/32 > 1/40`
`…………….`
`1/39 > 1/40`
`1/41 > 1/50`
`1/42 > 1/50`
`……………..`
`1/49 > 1/50`
`=> 1/26 + 1/27 + … + 1/50 > 5 . 1/30 + 10 . 1/40 + 10 . 1/50 = 1/6 + 1/4 + 1/5 = 37/60 > 36/60 = 3/5`
`=> 49/50 – (1/26 + 1/27 + … + 1/50) < 49/50 – 3/5 = 19/50 < 20/50 = 2/5`
$\text{Ta có : }$
`1/26 < 1/25`
`1/27 < 1/25`
`……………`
`1/30 < 1/25`
`1/31 < 1/30`
`1/32 < 1/30`
`……………`
`1/40 < 1/30`
`1/41 < 1/40`
`1/42 < 1/40`
`…………..`
`1/50 < 1/40`
`=> 1/26 +1/27 + … + 1/50 < 5 . 1/25 + 10 . 1/30 + 10 . 1/40 = 47/40 < 48/60 = 4/5`
`=> 49/50 – (1/26 + 1/27 + … + 1/50) > 49/50 – 4/5 = 9/50 > 10/50 = 1/5.`
$\text{Vậy }$`1/5 < A < 2/5.`