chứng tỏ rằng 1+1/2^2+1/3^2+1/4^2+…+1/99^2+1/100^2<9/5 27/07/2021 Bởi Ayla chứng tỏ rằng 1+1/2^2+1/3^2+1/4^2+…+1/99^2+1/100^2<9/5
Đáp án: `=>A<9/5 ( đpcm)` Giải thích các bước giải: Đặt `A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 < 9/5 ` `A =1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 < 1 + 1/(2.2) + 1/(3.3 ) + 1/(4.4) +….+1/(99.99) + 1/(100.100) < 9/5 ` `A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 < 1+ 1/4 + 1/(2.3) + 1/(3.4) +…+1/(98.99) + 1/(99.100)<9/5` `A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 < 5/4 + 1/2 – 1/3 – 1/4 +…+ 1/98 + 1/99 + 1/99 – 1/100 <9/5 ` `A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2<5/4 + 1/2 – 1/100 < 9/5 ` `A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 <5/4 + 49/100 < 9/5 ` `A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 <87/50 < 9/5 ` DO ` 9/5 = 90/50` Mà ` 87/50 < 90/50` `=>A<9/5 ( đpcm)` Bình luận
Đáp án: `A<9/5` Giải thích các bước giải: `A=1+1/2^2+1/3^2+1/4^2+…+1/99^2+1/100^2` `=>A=1+1/2.2+1/3.3+1/4.4+…+1/99.99+1/100.100` `=>A=1+1/4+1/3.3+1/4.4+…+1/99.99+1/100.100` `=>A<1+1/4+1/2.3+1/3.4+…+1/98.99+1/99.100` `=>A<5/4+1/2-1/3-1/4+…+1/98+1/99+1/99-1/100` `=>A<5/4+1/2-1/100` `=>A<5/4+49/100` `=>A<87/50<90/50=9/5` `=>A<9/5` Vậy `A<9/5`. Bình luận
Đáp án:
`=>A<9/5 ( đpcm)`
Giải thích các bước giải:
Đặt `A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 < 9/5 `
`A =1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 < 1 + 1/(2.2) + 1/(3.3 ) + 1/(4.4) +….+1/(99.99) + 1/(100.100) < 9/5 `
`A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 < 1+ 1/4 + 1/(2.3) + 1/(3.4) +…+1/(98.99) + 1/(99.100)<9/5`
`A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 < 5/4 + 1/2 – 1/3 – 1/4 +…+ 1/98 + 1/99 + 1/99 – 1/100 <9/5 `
`A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2<5/4 + 1/2 – 1/100 < 9/5 `
`A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 <5/4 + 49/100 < 9/5 `
`A=1 + 1/2^2 + 1/3^2 + 1/4^2 +…+1/99^2 + 1/100^2 <87/50 < 9/5 `
DO ` 9/5 = 90/50`
Mà ` 87/50 < 90/50`
`=>A<9/5 ( đpcm)`
Đáp án:
`A<9/5`
Giải thích các bước giải:
`A=1+1/2^2+1/3^2+1/4^2+…+1/99^2+1/100^2`
`=>A=1+1/2.2+1/3.3+1/4.4+…+1/99.99+1/100.100`
`=>A=1+1/4+1/3.3+1/4.4+…+1/99.99+1/100.100`
`=>A<1+1/4+1/2.3+1/3.4+…+1/98.99+1/99.100`
`=>A<5/4+1/2-1/3-1/4+…+1/98+1/99+1/99-1/100`
`=>A<5/4+1/2-1/100`
`=>A<5/4+49/100`
`=>A<87/50<90/50=9/5`
`=>A<9/5`
Vậy `A<9/5`.