CMR 1/3-2/3^2+3/3^3-4/3^4+……….+99/3^99-100/3^100<3/4. Giải ngắn gọn

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1. Đáp án:

   Giải thích các bước giải:

   Đặt C = 1/3^1 + 2/3^2 + .3/3^3 + .. + 100/3^100 
   1/3*C = 1/3^2 + 2/3^3 + 3/3^4 + .. + 100/3^101 
   => C – 1/3*C = 1/3^1 + (2/3^2 – 1/3^2) + (3/3^3 – 2/3^3) + .. + (100/3^100 – 99/3^100) – 100/3^101 
   => 2/3*C = 1/3^1 + 1/3^2 + 1/3^3 + .. + 1/3^100 – 100/3^101 
   + xét S= 1/3^1 + 1/3^2 + 1/3^3 + .. + 1/3^100 tương tự 
   1/3*S = 1/3^2 + 1/3^3 + 1/3^4 + .. + 1/3^101 
   => S – 1/3*S = 1/3^1 – 1/3^101 
   <=> 2/3*S = (1/3 – 1/3^101) 
   <=> S = 3/2*(1/3 – 1/3^101) thay vào C ta có 
   2/3*C = 3/2*(1/3 – 1/3^101) – 100/3^101 
   <=> C = 9/4*(1/3 – 1/3^101) – 150/3^101 
   <=>C = 3/4 – 9/4*1/3^101 – 150/3^101 < 3/4 =>ĐPCM

             *CHO MÌNH HAY NHẤT NHA*

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