\(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+..........+\dfrac{1}{256}+\dfrac{1}{512}=?\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{8}+...+\dfrac{1}{256}-\dfrac{1}{512}-\dfrac{1}{512}\)
\(=1-\dfrac{1}{512}\)
\(=\dfrac{511}{512}\)
Vậy \(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+.........+\dfrac{1}{256}+\dfrac{1}{512}=\dfrac{511}{512}\)
Bài bạn trên cách trình bày mk ko hiểu lắm! mk làm lại nhé!
Đặt :
\(S=\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+...........+\dfrac{1}{256}+\dfrac{1}{512}\)
\(\Leftrightarrow S=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+.........+\dfrac{1}{2^8}+\dfrac{1}{2^9}\)
\(\Leftrightarrow2S=2\left(\dfrac{1}{2}+\dfrac{1}{2^2}+.........+\dfrac{1}{2^9}\right)\)
\(\Leftrightarrow2S=1+\dfrac{1}{2}+\dfrac{1}{2^2}+.........+\dfrac{1}{2^8}\)
\(\Leftrightarrow2S-S=\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+.....+\dfrac{1}{2^8}\right)-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+......+\dfrac{1}{2^9}\right)\)
\(\Leftrightarrow S=1-\dfrac{1}{2^9}\)
\(\Leftrightarrow S=1-\dfrac{1}{512}=\dfrac{511}{512}\)
Ta có : \(\dfrac{1}{2}\) = 1 - \(\dfrac{1}{2}\) ; \(\dfrac{1}{4}\) = \(\dfrac{1}{2}\) - \(\dfrac{1}{4}\) ; \(\dfrac{1}{8}\) = \(\dfrac{1}{4}\) - \(\dfrac{1}{8}\) ;............. ;
\(\dfrac{1}{512}\) = \(\dfrac{1}{256}\) - \(\dfrac{1}{512}\)
Vậy : \(\dfrac{1}{2}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{8}\) + ................... + \(\dfrac{1}{256}\) + \(\dfrac{1}{512}\)
= 1 - \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) - \(\dfrac{1}{4}\) + \(\dfrac{1}{4}\) - \(\dfrac{1}{8}\) + ............. + \(\dfrac{1}{256}\) - \(\dfrac{1}{512}\)
= 1 - \(\dfrac{1}{512}\)
= \(\dfrac{511}{512}\)
