cho S = 1/5 mũ 2+ 1/6 mũ 2+……..+1/2020 mũ 2 21/11/2021 Bởi Cora cho S = 1/5 mũ 2+ 1/6 mũ 2+……..+1/2020 mũ 2
Ta có: ` 1/5^2 < \frac{1}{4.5} ` ` 1/6^2 < \frac{1}{5.6} ` ` ……. ` ` 1/2020^2 < \frac{1}{2019.2020} ` ` => 1/5^2 + 1/6^2 + … + 1/2020^2 < \frac{1}{4.5} + \frac{1}{5.6} + … + \frac{1}{2019.2020} ` ` => S < \frac{1}{4.5} + \frac{1}{5.6} + … + \frac{1}{2019.2020} ` ` => S < 1/4 – 1/5 + 1/5 – 1/6 + … + 1/2019 . 1/2020 ` ` => S < 1/4 – 1/2020 ` Do: ` 1/4 – 1/2020 < 1/4 ` ` => S < 1/4 ` (1) Ta có: ` 1/5^2 > \frac{1}{5.6} ` ` 1/6^2 > \frac{1}{6.7} ` ` ……….. ` ` 1/2020^2 > \frac{1}{2020.2021} ` ` => 1/5^2 + 1/6^2 + … + 1/2020^2 > \frac{1}{5.6} + \frac{1}{6.7} + … + \frac{1}{2020.2021} ` ` => S > \frac{1}{5.6} + \frac{1}{6.7} + … + \frac{1}{2020.2021} ` ` => S > 1/5 – 1/6 + 1/6 – 1/7 + … + 1/2020 – 1/2021 ` ` => S > 1/5 – 1/2021 ` Do: ` 1/5 – 2021 > 1/6 ` ` => S > 1/6 ` (2) Từ (1) và (2): ` => 1/6 < S < 1/4 ` ` (đpcm) ` Bình luận
Cách làm ta thấy 1/5^2<1/4.5 1/6^2<1/5.6 …………………… 1/2020^2<1/2019.2020 => 1/5^2+1/6^2+…+1/2020^2<1/4.5+1/5.6+1/2019.2020 =1/4-1/5+1/5-1/6+…+1/2019-1/2020 =1/4-1/2020 lại có 1/5^2>1/5.6 1/6^2>1/6.7 ……………………. 1/2020^2>1/2020.2021 mk dang bận nên chỉ làm đc đến đây thôi chúc bạn học tốt Bình luận
Ta có:
` 1/5^2 < \frac{1}{4.5} `
` 1/6^2 < \frac{1}{5.6} `
` ……. `
` 1/2020^2 < \frac{1}{2019.2020} `
` => 1/5^2 + 1/6^2 + … + 1/2020^2 < \frac{1}{4.5} + \frac{1}{5.6} + … + \frac{1}{2019.2020} `
` => S < \frac{1}{4.5} + \frac{1}{5.6} + … + \frac{1}{2019.2020} `
` => S < 1/4 – 1/5 + 1/5 – 1/6 + … + 1/2019 . 1/2020 `
` => S < 1/4 – 1/2020 `
Do: ` 1/4 – 1/2020 < 1/4 `
` => S < 1/4 ` (1)
Ta có:
` 1/5^2 > \frac{1}{5.6} `
` 1/6^2 > \frac{1}{6.7} `
` ……….. `
` 1/2020^2 > \frac{1}{2020.2021} `
` => 1/5^2 + 1/6^2 + … + 1/2020^2 > \frac{1}{5.6} + \frac{1}{6.7} + … + \frac{1}{2020.2021} `
` => S > \frac{1}{5.6} + \frac{1}{6.7} + … + \frac{1}{2020.2021} `
` => S > 1/5 – 1/6 + 1/6 – 1/7 + … + 1/2020 – 1/2021 `
` => S > 1/5 – 1/2021 `
Do: ` 1/5 – 2021 > 1/6 `
` => S > 1/6 ` (2)
Từ (1) và (2):
` => 1/6 < S < 1/4 ` ` (đpcm) `
Cách làm
ta thấy
1/5^2<1/4.5
1/6^2<1/5.6
……………………
1/2020^2<1/2019.2020
=> 1/5^2+1/6^2+…+1/2020^2<1/4.5+1/5.6+1/2019.2020
=1/4-1/5+1/5-1/6+…+1/2019-1/2020
=1/4-1/2020
lại có
1/5^2>1/5.6
1/6^2>1/6.7
…………………….
1/2020^2>1/2020.2021
mk dang bận nên chỉ làm đc đến đây thôi chúc bạn học tốt