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 => dpcm