Chứng minh:`A=1/4+1/16+1/36+1/64+1/100+1/144+1/196+12/256+1/324<1/2` 14/08/2021 Bởi Faith Chứng minh:`A=1/4+1/16+1/36+1/64+1/100+1/144+1/196+12/256+1/324<1/2`
Đáp án: `A<1/2` Giải thích các bước giải: đề lỗi ở `12/256` thành `1/256` ta có : `A=1/4+1/16+1/36+1/64+1/100+1/144+1/196+12/256+1/324<1/2` `A=1/4+1/16+1/36+1/64+1/100+1/144+1/196+12/256+1/324` `=1/2^2+1/4^2+1/6^2+1/8^2+1/10^2+1/12^2+1/14^2+1/16^2+1/18^2` do : `1/2^2<1/1.3` `1/4^2<1/3.5` `1/6^2<1/5.7` `. . . . . ` `1/18^2<1/17.19` ______________________________________________________ `=>` `A<1/1.3+1/3.5+1/5.7+…+1/17.19` `=>` `1/2×(1-1/3+1/3-1/5+1/5-1/7+…+1/17-1/19)` `=>` `1/2×(1-1/19)` `=>` `1/2×18/19` `=>` `9/19` do `9/19` `<` `1/2` nên `A<1/2` Bình luận
A= 1/4+1/16+1/36+1/64+1/100+1/144+1/196+1/256+1/324+1/400. A = 1/4.(1+1/2^2+1/3^2+1/4^2+1/5^2+1/6^2+1/7^2+1/8^2+1/9^2) <1/4.(1+1/1.2+1/2.3+1/3.4+1/4.5+1/5.6+1/6.7+1/7.8+1/8.9+1/9.10) =1/4.(1+1-1/2+1/2-1/3+…+1/8-1/9+1/9-1/10) =19/40<20/40=1/2 Bình luận
Đáp án:
`A<1/2`
Giải thích các bước giải:
đề lỗi ở `12/256` thành `1/256`
ta có : `A=1/4+1/16+1/36+1/64+1/100+1/144+1/196+12/256+1/324<1/2`
`A=1/4+1/16+1/36+1/64+1/100+1/144+1/196+12/256+1/324`
`=1/2^2+1/4^2+1/6^2+1/8^2+1/10^2+1/12^2+1/14^2+1/16^2+1/18^2`
do :
`1/2^2<1/1.3`
`1/4^2<1/3.5`
`1/6^2<1/5.7`
`. . . . . `
`1/18^2<1/17.19`
______________________________________________________
`=>` `A<1/1.3+1/3.5+1/5.7+…+1/17.19`
`=>` `1/2×(1-1/3+1/3-1/5+1/5-1/7+…+1/17-1/19)`
`=>` `1/2×(1-1/19)`
`=>` `1/2×18/19`
`=>` `9/19`
do `9/19` `<` `1/2`
nên `A<1/2`
A= 1/4+1/16+1/36+1/64+1/100+1/144+1/196+1/256+1/324+1/400.
A = 1/4.(1+1/2^2+1/3^2+1/4^2+1/5^2+1/6^2+1/7^2+1/8^2+1/9^2) <1/4.(1+1/1.2+1/2.3+1/3.4+1/4.5+1/5.6+1/6.7+1/7.8+1/8.9+1/9.10)
=1/4.(1+1-1/2+1/2-1/3+…+1/8-1/9+1/9-1/10)
=19/40<20/40=1/2