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![](https://rs.olm.vn/images/avt/0.png?1311)
Nhận xét :
1/2 = 1 - 1/2 ; 1/4 = 1/2 - 1/4 ; 1/8 = 1/4 - 1/8 ; ..... ; 1/256 = 1/128 - 1/256
=> A = 1 - 1/2 + 1/2 - 1/4 + 1/4 - 1/8 + ..... + 1/128 - 1/256
=> A = 1 - 1/256 = 255/256
![](https://rs.olm.vn/images/avt/0.png?1311)
quy đồngcác phân số lấy mẫu số là 512 .ta có tử số là
256 +128 + 64 +32 +16 +8 +4 +2 +1 =495
A =\(\frac{495}{512}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có:
2A=1+1/2+1/4+1/8+1/16+1/32+1/64
=> 2A-A=(1+1/2+1/4+1/8+1/16+1/32+1/64) - (1/2+1/4+1/8+1/16+1/32+1/64+1/128)
A= 1-1/128=127/128
\(\frac{1}{2}+\frac{1}{4}=1-\frac{1}{4}\)
\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}=1-\frac{1}{8}\)
............................................
\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{128}=1-\frac{1}{128}=\frac{127}{128}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
= 1 - 1/2+ 1/2- 1/4 +1/4 - 1/8 +1/8 -1/16 +1/16 -1/32 +1/32 -1/64 +1/64 - 1/128
= 1-1/128
=127/128
\(\frac{1}{2}\)+ \(\frac{1}{4}\)+ \(\frac{1}{8}\)+ \(\frac{1}{16}\)+ \(\frac{1}{32}\)+ \(\frac{1}{64}\)+ \(\frac{1}{128}\)= \(\frac{64}{128}\)+ \(\frac{32}{128}\)+ \(\frac{16}{128}\)+ \(\frac{8}{128}\)+ \(\frac{4}{128}\)+ \(\frac{2}{128}\)+ \(\frac{1}{128}\).
= \(\frac{127}{128}\).
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{1}{2}\)+ \(\frac{1}{4}\)+ \(\frac{1}{8}\)+ \(\frac{1}{16}\)+ \(\frac{1}{32}\)+ \(\frac{1}{64}\)+ \(\frac{1}{128}\)
= \(1\)- \(\frac{1}{2}\)+ \(\frac{1}{2}\)- \(\frac{1}{4}\)+ \(\frac{1}{4}\)- \(\frac{1}{8}\)+ \(\frac{1}{8}\)- \(\frac{1}{16}\)+ \(\frac{1}{16}\)- \(\frac{1}{32}\)+ \(\frac{1}{32}\)- \(\frac{1}{64}\)+ \(\frac{1}{64}\)- \(\frac{1}{128}\)
= \(1\)- \(\frac{1}{128}\)
= \(\frac{127}{128}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{256}\)
=\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{8}+...+\frac{1}{128}-\frac{1}{256}\)
=\(1-\frac{1}{256}\)
=\(\frac{255}{256}\)
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256
= 128/256 + 64/256 + 32/256 + 16/256 + 8/256 + 4/256 + 2/128 + 1/256
= 255/256
\(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}\)
\(\Rightarrow2A=\frac{2}{2}+\frac{2}{4}+\frac{2}{8}+\frac{2}{16}+\frac{2}{32}+\frac{2}{64}+\frac{2}{128}\)
\(\Rightarrow2A=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}\)
\(\Rightarrow2A-A=\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}\right)\)
\(\Rightarrow A=1-\frac{1}{128}=\frac{128}{128}-\frac{1}{128}=\frac{127}{128}\)
\(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....+\frac{1}{128}\)
\(=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{8}\right)+.....+\left(\frac{1}{64}-\frac{1}{128}\right)\)
\(=1-\frac{1}{128}=\frac{127}{128}\)