So sánh : $\frac{2018^{99} -1 }{2018^{100} -1 }$ và $\frac{$2018^{98} -1 }{$2018^{99} -1 }$ 28/08/2021 Bởi Aubrey So sánh : $\frac{2018^{99} -1 }{2018^{100} -1 }$ và $\frac{$2018^{98} -1 }{$2018^{99} -1 }$
Gọi (2018^99-1/2018^100-1) là A; (2018^98-1/2018^99-1) là B Ta có: A = (2018^99-1/2018^100-1) => 2018.A=(2018^100-2018)/(2018^100-1) => 2018.A=1-(2017/2018^100-1) B = (2018^98-1/2018^99-1) => 2018.B=(2018^99-2018)/(2018^99-1) => 2018.B=1-(2017/2018^99-1) Vì (2017/2018^100-1)<(2017/2018^99-1) => 1-(2017/2018^100-1)>1-(2017/2018^99-1) Vậy A>B Bình luận
Đáp án: `\frac{2018^{99} -1 }{2018^{100} -1 }>\frac{2018^{98} -1 }{2018^{99} -1 }` Giải thích các bước giải: Đặt `A=\frac{2018^{99} -1 }{2018^{100} -1 }` và `B=\frac{2018^{98} -1 }{2018^{99} -1 }` `A=\frac{2018^{99} -1 }{2018^{100} -1 }` `=>2018A={2018(2018^{99} -1 )}/{2018^{100} -1 }` `=>2018A={2018^{100} -2018}/{2018^{100} -1 }` `=>2018A={(2018^{100}-1) -2017}/{2018^{100} -1 }` `=>2018A=1-{2017}/{2018^{100} -1 }(1)` `B=\frac{2018^{98} -1 }{2018^{99} -1 }` `=>2018B={2018(2018^{98} -1 )}/{2018^{99} -1 }` `=>2018B={2018^{99} -2018 }/{2018^{99} -1 }` `=>2018B={2018^{99} -2018 }/{2018^{99} -1 }` `=>2018B={(2018^{99} -1)-2017 }/{2018^{99} -1 }` `=>2018B=1-{2017 }/{2018^{99} -1 }(2)` Vì `2018^{99} -1 <2018^{100} -1` `=>{2017}/{2018^{100} -1 }<{2017 }/{2018^{99} -1 }(3)` Từ `(1),(2),(3)` `=>2018A>2018B` `=>A>B` Bình luận
Gọi (2018^99-1/2018^100-1) là A; (2018^98-1/2018^99-1) là B
Ta có: A = (2018^99-1/2018^100-1) => 2018.A=(2018^100-2018)/(2018^100-1)
=> 2018.A=1-(2017/2018^100-1)
B = (2018^98-1/2018^99-1) => 2018.B=(2018^99-2018)/(2018^99-1)
=> 2018.B=1-(2017/2018^99-1)
Vì (2017/2018^100-1)<(2017/2018^99-1) => 1-(2017/2018^100-1)>1-(2017/2018^99-1)
Vậy A>B
Đáp án:
`\frac{2018^{99} -1 }{2018^{100} -1 }>\frac{2018^{98} -1 }{2018^{99} -1 }`
Giải thích các bước giải:
Đặt `A=\frac{2018^{99} -1 }{2018^{100} -1 }` và `B=\frac{2018^{98} -1 }{2018^{99} -1 }`
`A=\frac{2018^{99} -1 }{2018^{100} -1 }`
`=>2018A={2018(2018^{99} -1 )}/{2018^{100} -1 }`
`=>2018A={2018^{100} -2018}/{2018^{100} -1 }`
`=>2018A={(2018^{100}-1) -2017}/{2018^{100} -1 }`
`=>2018A=1-{2017}/{2018^{100} -1 }(1)`
`B=\frac{2018^{98} -1 }{2018^{99} -1 }`
`=>2018B={2018(2018^{98} -1 )}/{2018^{99} -1 }`
`=>2018B={2018^{99} -2018 }/{2018^{99} -1 }`
`=>2018B={2018^{99} -2018 }/{2018^{99} -1 }`
`=>2018B={(2018^{99} -1)-2017 }/{2018^{99} -1 }`
`=>2018B=1-{2017 }/{2018^{99} -1 }(2)`
Vì `2018^{99} -1 <2018^{100} -1`
`=>{2017}/{2018^{100} -1 }<{2017 }/{2018^{99} -1 }(3)`
Từ `(1),(2),(3)`
`=>2018A>2018B`
`=>A>B`