Bài 7. Cho M = (1/1+ 1/2+ 1/3+ 1/4+⋯+ 1/2018).2.3.4…2018
Chứng minh M chia hết cho 2019
Bài 8. Cho A = 1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + 1/128 – 1/156 . Chứng minh A < 1/3
Bài 7. Cho M = (1/1+ 1/2+ 1/3+ 1/4+⋯+ 1/2018).2.3.4…2018 Chứng minh M chia hết cho 2019 Bài 8. Cho A = 1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + 1/128 –
By Hailey
Bài 7)
Ta có: M = ($\frac{1}{1}$ + $\frac{1}{2}$ + $\frac{1}{3}$ + $\frac{1}{4}$ +…+ $\frac{1}{2018}$).2.3.4…2018
⇒ M = ($\frac{1}{1}$ + $\frac{1}{2}$ + $\frac{1}{3}$ + $\frac{1}{4}$ +…+ $\frac{1}{2018}$).(673.3).2.4….2018
⇒ M = ($\frac{1}{1}$ + $\frac{1}{2}$ + $\frac{1}{3}$ + $\frac{1}{4}$ +…+ $\frac{1}{2018}$).2019.2.4….2018
Vì M có một thừa số 2019 nên 2019 nhân với bất cứ số nào cũng chia hết cho 2019 nên M chia hết cho 2019
vậy M chia hết cho 2019 (đpcm)
Đáp án:
Bài 7:
Ta có:
`M = (1 + 1/2 + 1/3 + 1/4 + …………….. + 1/2018) . 2 . 3 . 4 ……..2018`
`M =[ (1 + 1/2018) + (1/2 +1/2017) + (1/3 + 1/2016) + ……………… +(1/1009 + 1/1010)] . 2 . 3 . 4 . …. 2018`
`M=[ 2019/2018+ 2019/(2.2017) + 2019/(3 . 2016) + ……. + 2009/(1009.1010) ] . 2 . 3 .4 … 2018`
`M =( 1/2018 + 1/(2.2017) + 1/( 3 . 2016) + …………. + 1/(1009 . 1010) ) . 2019 . 2 . 3 . 4 … 2018 ⋮ 2019`
Bài 8:
Bạn ghi sai đề nên mình sửa
`A = 1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + 1/128 – 1/256`
`2A = 2 . (1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + 1/128 – 1/256)`
`2A = 1 – 1/2 + 1/4 – 1/8 + 1/16 – 1/32 + 1/64 – 1/128`
`2A + A = (1 – 1/2 + 1/4 – 1/8 + 1/16 – 1/32 + 1/64 – 1/128) +(1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + 1/128 – 1/256)`
`2A + A = 1 – 1/2 + 1/4 – 1/8 + 1/16 – 1/32 + 1/64 – 1/128 + 1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + 1/128 – 1/256`
`3A = 1 – (1/2 – 1/2) + (1/4 – 1/4) – (1/8 – 1/8) + ………… + (1/64 – 1/64) – (1/128 – 1/128) – 1/256`
`3A = 1 – 1/256 < 1`
`⇒ A < 1/3`
`text{ @toanisthebest}`