Bài 1: a) chứng minh rằng: 1 phần 2²+1 phần 3²+1 phần 4²+…+1 phần 100² <3 phần 4 27/10/2021 Bởi Ximena Bài 1: a) chứng minh rằng: 1 phần 2²+1 phần 3²+1 phần 4²+…+1 phần 100² <3 phần 4
$\text{ Đặt A=}$${\dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2}+…+\dfrac{1}{100^2}}$$\text{ Ta thấy}$${ \dfrac{1}{2^2}=\dfrac{1}{2.2}=\dfrac{1}{4}}$${ \dfrac{1}{3^2}=\dfrac{1}{3.3} < \dfrac{1}{2.3}}$${ \dfrac{1}{4^2}=\dfrac{1}{4.4} < \dfrac{1}{3.4}}$….${\dfrac{1}{100^2}=\dfrac{1}{100.100} < \dfrac{1}{99.100}}$ ⇒ ${\dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2}+…+\dfrac{1}{100^2}}$ < ${\dfrac{1}{4}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+…+\dfrac{1}{99.100}}$ ⇒ ${A}$ < ${\dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+…+\dfrac{1}{99}+\dfrac{1}{100}}$ ⇒ ${A}$ ${< \dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{100}}$ ⇒ ${A}$ ${< \dfrac{3}{4} – \dfrac{1}{100}}$ ⇒ ${A}$${ < \dfrac{3}{4}}$ Bình luận
Đặt `H = 1/2^2 + 1/3^2 + 1/4^2 + … + 1/100^2` Ta có: `1/2^2 = 1/2.2 = 1/4` `1/3^2 = 1/3.3 < 1/2.3` `1/4^2 = 1/4.4 < 1/3.4` … `1/100^2 = 1/100.100 < 1/99.100` `=> H < 1/4 + 1/2.3 + 1/3.4 + … + 1/99.100` `=> H < 1/4 + 1/2 – 1/3 + 1/3 – 1/4 + … + 1/99 – 1/100` `=> H < 1/4 + 1/2 – 1/100` `=> H < 3/4 – 1/100 < 3/4` `=> H < 3/4` (đpcm) Chúc bạn học tốt! Bình luận
$\text{ Đặt A=}$${\dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2}+…+\dfrac{1}{100^2}}$
$\text{ Ta thấy}$
${ \dfrac{1}{2^2}=\dfrac{1}{2.2}=\dfrac{1}{4}}$
${ \dfrac{1}{3^2}=\dfrac{1}{3.3} < \dfrac{1}{2.3}}$
${ \dfrac{1}{4^2}=\dfrac{1}{4.4} < \dfrac{1}{3.4}}$
….
${\dfrac{1}{100^2}=\dfrac{1}{100.100} < \dfrac{1}{99.100}}$
⇒ ${\dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2}+…+\dfrac{1}{100^2}}$ < ${\dfrac{1}{4}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+…+\dfrac{1}{99.100}}$
⇒ ${A}$ < ${\dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+…+\dfrac{1}{99}+\dfrac{1}{100}}$
⇒ ${A}$ ${< \dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{100}}$
⇒ ${A}$ ${< \dfrac{3}{4} – \dfrac{1}{100}}$
⇒ ${A}$${ < \dfrac{3}{4}}$
Đặt `H = 1/2^2 + 1/3^2 + 1/4^2 + … + 1/100^2`
Ta có: `1/2^2 = 1/2.2 = 1/4`
`1/3^2 = 1/3.3 < 1/2.3`
`1/4^2 = 1/4.4 < 1/3.4`
…
`1/100^2 = 1/100.100 < 1/99.100`
`=> H < 1/4 + 1/2.3 + 1/3.4 + … + 1/99.100`
`=> H < 1/4 + 1/2 – 1/3 + 1/3 – 1/4 + … + 1/99 – 1/100`
`=> H < 1/4 + 1/2 – 1/100`
`=> H < 3/4 – 1/100 < 3/4`
`=> H < 3/4` (đpcm)
Chúc bạn học tốt!