$\frac{2}{3}$ + $\frac{2}{3^2} $ + $\frac{2}{3^3}$ + $\frac{2}{3^4}$ + ….+ $\frac{2}{3 ^2012}$

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1. Đáp án:

    `1-1/3^2012`

   Giải thích các bước giải:

   `2/3+2/3^2+2/3^3+…+2/3^2012`

   `=2.(1/3+1/3^2+1/3^3+…+1/3^2012)`

   Đặt `A= 1/3+1/3^2+1/3^3+…+1/3^2012`

   `3A=1+1/3+1/3^2+…+1/3^2011`

   `3A-A=2A=1-1/3^2012`

   `A=(1-1/3^2012)/2`

   `=2.(1-1/3^2012)/2=1-1/3^2012`

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2. `A = 2/3 + 2/3^2 + 2/3^3 + … + 2/3^2018`

   `⇒ 3A = 2 + 2/3 + 2/3^2 + … + 2/3^2017`

   `⇒ 3A – A = (2 + 2/3 + … + 2/3^2017) – (2/3 + 2/3^2 + … + 2/3^2018)`

   `⇒ 3A – A = 2 – 2/3^2018`

   `⇒ 2A = 2 – 2/3^2018`

   `⇒ A = (2 – 2/3^2018) : 2`

    

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