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Đặt \(A=1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-..-\frac{1}{2048}\)
\(\Rightarrow A=1-\left(1-\frac{1}{2}\right)-\left(\frac{1}{2}-\frac{1}{4}\right)-..-\left(\frac{1}{1024}-\frac{1}{2048}\right)\)
\(\Rightarrow A=1-1+\frac{1}{2}-\frac{1}{2}+\frac{1}{4}-..-\frac{1}{1024}+\frac{1}{2018}\)
\(\Rightarrow A+\frac{1}{2018}\)
1-1/2-1/4-1/8-1/16-1/32-1/64-1/128-1/256-1/512-1/1024-1/2048 =0.00048828125
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta thấy các mẫu số đều là số chẵn và là bội số của 2
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/1024
Ta có :
1/512 = 1×2 / 512×2 = 2/1024
1/256 = 1×4 / 256×4 = 4/1024
1/128 = 1×8 / 128×8 = 8/1024
1/64 = 1×16 / 64×16 = 16/1024
1/32 = 1×32 / 32×32 = 32/1024
1/16 = 1×64 / 16×64 = 64/1024
1/8 = 1×128 / 8×128 = 128/1024
1/4 = 1×256 / 4×256 = 256/1024
1/2 = 1×512 / 2×512 = 512/1024
=> 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/1024
= 512/1024 + 256/1024 + 128/1024 + 64/1024 + 32/1024 + 16/1024 + 8/1024 + 4/1024 + 2/1024 + 1/1024
= (512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1) / 1024
= 1023/1024
Tick nha
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A = 1/2 + 1/4 + 1/8 + ... + 1/1024
2A = 1 + 1/2 + 1/4 + ... + 1/512
2A - A = (1 + 1/2 + 1/4 + ... + 1/512) - (1/2 + 1/4 + 1/8 + ... + 1/1024)
A = 1 - 1/1024
A = 1023/1024
\(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....+\frac{1}{1024}\)
\(\Rightarrow2A=1+\frac{1}{2}+\frac{1}{4}+......+\frac{1}{512}\)
\(\Rightarrow A=2A-A=1-\frac{1}{1024}\)
\(A=\frac{1023}{1024}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
2A=1+1/2+1/4+1/8+.........+1/512
2A-A=(1+1/2+1/4+1/8+....+1/512)-(1/2+1/4+1/8+.....+1/1024)
A=1-1/1024
A=1023/1024
vậy A=1023/1024
A = (1/2+1/64)+(1/4+1/128)+(1/8+1/256)+(1/16+1/512)+(1/32+1/1024)
= 1/2(1+1/32)+1/4(1+1/32)+1/8(1+1/32)+1/16(1+1/32)+1/32(1+1/32)
= (1+1/32) x (1/2+/14+1/8+1/16+1/32)
= (33/32) x (31/32) = 1023/1024
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt A = 1/2 + 1/4 + 1/8 + ... + 1/1024
2A = 1 + 1/2 + 1/4 + ... + 1/512
2A - A = (1 + 1/2 + 1/4 + ... + 1/512) - (1/2 + 1/4 + 1/8 + ... + 1/1024)
A = 1 - 1/1024
A = 1023/1024
![](https://rs.olm.vn/images/avt/0.png?1311)
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![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\frac{3}{16}+\frac{3}{32}+\frac{3}{64}+\frac{3}{128}+\frac{3}{256}+\frac{3}{512}+\frac{3}{1024}\)
=\(3.\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}+\frac{1}{512}+\frac{1}{1024}\right)\)
=\(3.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{8}+\frac{1}{8}-\frac{1}{16}+\frac{1}{16}-\frac{1}{32}+\frac{1}{32}-\frac{1}{64}+\frac{1}{64}-\frac{1}{128}+\frac{1}{128}-\frac{1}{256}+\frac{1}{256}-\frac{1}{512}+\frac{1}{512}-\frac{1}{1024}\right)\)
=\(3.\left(1-\frac{1}{1024}\right)=3.\left(\frac{1024}{1024}-\frac{1}{1024}\right)=3.\frac{1023}{1024}=\frac{3069}{1024}\)
Chúc em học tốt
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Đặt A × 2 = 2 + 4 +8 +16 + 32 + ....+ 1024+2048
Cùng thêm 1 và bớt 1 ta có như sau:
A × 2 = 1 + 2 + 4 + 8 + 16 + .....+1024+2048 -1
A × 2 = A +2048 - 1
A = 2048 -1 =2047
đặt 1+2+4+8+16+32+...+1024 là A
theo bài ra ta có
2A=2+4+8+16+32+64+...+2048
-
A=1+2+4+8+16+32+...+1024
A=2048-1=2047
Cách này có vẻ hơi khó hiểu nhưng chỉ cần bạn tính 2A-A ra là đc