1) a. Cho A= $\frac{1}{2^2}$ + $\frac{1}{3^2}$ + ……+ $\frac{1}{2012^2}$ + $\frac{1}{2013^2}$ . Hãy chứng tỏ rẳng A < 1. b. Cho A = $\frac{2}{3}$ + $\frac{2}{3^2}$ + $\frac{2}{3^3}$ +.....+ $\frac{2}{3^2018}$ . Hãy chứng tỏ rằng A < 1. 2) Tìm số tự nhiên x biết rằng: $\frac{1}{3}$ + $\frac{1}{6}$ + $\frac{1}{10}$ +....+ $\frac{2}{x(x+1)}$ = $\frac{2015}{2017}$ Giúp mk vs mn ơi!
`1) a. A = 1/2^2 + 1/3^2 + … + 1/2012^2 + 1/2013^2`
`< 1/1.2 + 1/2.3 + … + 1/2011.2012 + 1/2012.2013`
`= 1 – 1/2 + 1/2 – 1/3 + …+ 1/2011 – 1/2012 + 1/2012 – 1/2013`
`= 1 – 1/2013 < 1`
` Vậy ` `A < 1 (đpcm)`
`b. A = 2/3 + 2/3^2 + 2/3^3 +…+ 2/3^2018`
`⇒ 1/2 A = 1/3 + 1/3^2 + 1/3^3 +…+ 1/3^2018`
`⇒ 1/6 A = 1/3^2 + 1/3^3 + 1/3^4 +…+ 1/3^2019`
`⇒ 1/2 A – 1/6 A = (1/3 + 1/3^2 + 1/3^3 +…+ 1/3^2018) – (1/3^2 + 1/3^3 + 1/3^4 +…+ 1/3^2019)`
`⇒ 1/3 A = 1/3 – 1/3^2019`
`⇒ A = (1/3 – 1/3^2019) . 3`
`⇒ A = 1 – 1/3^2018 < 1`
`2) 1/3 + 1/6 + 1/10 +….+ 2/(x.(x+1)) = 2015/2017`
`⇒ 2 . (1/6 + 1/12 + 1/20 +….+ 1/(x.(x+1))) = 2015/2017`
`⇒ 1/2.3 + 1/3.4 + 1/4.5 +….+ 1/(x.(x+1)) = 2015/2017 : 2“
`⇒ 1/2 – 1/3 + 1/3 – 1/4 + 1/4 – 1/5 +… + 1/x – 1/(x+1) = 2015/4034`
`⇒ 1/2 – 1/(x+1) = 2015/4034`
`⇒ 1/(x+1) = 1/2 – 2015/4034`
`⇒ 1/(x+1) = 1/2017`
`⇒ x + 1 = 2017`
`⇒ x = 2016`