Toán chứng tỏ S=1/2.2+1/2.2.2+1/2.2.2.2+…+1/2.mũ 100<1 28/07/2021 By Reese chứng tỏ S=1/2.2+1/2.2.2+1/2.2.2.2+…+1/2.mũ 100<1
`S= 1/2^2 + 1/2^3 + 1/2^4 +….+ 1/2^100` `1/2 S = 1/2( 1/2^2 + 1/2^3 + 1/2^4 +…+ 1/2^100)` `1/2 S= 1/2^3 + 1/2^4 + 1/2^5 +…+ 1/2^101` `S- 1/2S = 1/2^2+ 1/2^3 + 1/2^4 +…+ 1/2^100 – 1/2^3 – 1/2^4- 1/2^5 -…- 1/2^101` `1/2 S= 1/2^2 – 1/2^101` `S= (1/2^2 – 1/2^101) : 1/2` `S= (1/2^2 – 1/2^101) . 2` `S= 1/2 – 1/2^100` Vì `1/2< 1` `=> 1/2 – 1/2^100 < 1` Vậy `S < 1` Trả lời
`S = 1/(2.2) + 1/(2.2.2) + 1/(2.2.2.2) + … + 1/2^100` `⇒ S = 1/2^2 + 1/2^3 + 1/2^4 + … + 1/2^100``⇒ 2S = 2. (1/2^2 + 1/2^3 + 1/2^4 + … + 1/2^100)` `⇒ 2S = 1/2 + 1/2^2 + 1/2^3 + … + 1/2^99` `⇒ 2S – S = (1/2 + 1/2^2 + 1/2^3 + … + 1/2^99) – (1/2^2 + 1/2^3 + 1/2^4 + … + 1/2^100)` `⇒ S = 1/2 – 1/2^100` `Vì 1/2 < 1 ⇒ 1/2 – 1/2^100 < 1` `⇒ S < 1` `⇒ đpcm` Trả lời
`S= 1/2^2 + 1/2^3 + 1/2^4 +….+ 1/2^100`
`1/2 S = 1/2( 1/2^2 + 1/2^3 + 1/2^4 +…+ 1/2^100)`
`1/2 S= 1/2^3 + 1/2^4 + 1/2^5 +…+ 1/2^101`
`S- 1/2S = 1/2^2+ 1/2^3 + 1/2^4 +…+ 1/2^100 – 1/2^3 – 1/2^4- 1/2^5 -…- 1/2^101`
`1/2 S= 1/2^2 – 1/2^101`
`S= (1/2^2 – 1/2^101) : 1/2`
`S= (1/2^2 – 1/2^101) . 2`
`S= 1/2 – 1/2^100`
Vì `1/2< 1`
`=> 1/2 – 1/2^100 < 1`
Vậy `S < 1`
`S = 1/(2.2) + 1/(2.2.2) + 1/(2.2.2.2) + … + 1/2^100`
`⇒ S = 1/2^2 + 1/2^3 + 1/2^4 + … + 1/2^100`
`⇒ 2S = 2. (1/2^2 + 1/2^3 + 1/2^4 + … + 1/2^100)`
`⇒ 2S = 1/2 + 1/2^2 + 1/2^3 + … + 1/2^99`
`⇒ 2S – S = (1/2 + 1/2^2 + 1/2^3 + … + 1/2^99) – (1/2^2 + 1/2^3 + 1/2^4 + … + 1/2^100)`
`⇒ S = 1/2 – 1/2^100`
`Vì 1/2 < 1 ⇒ 1/2 – 1/2^100 < 1`
`⇒ S < 1`
`⇒ đpcm`