2

4

8

16

32

64

128

256

2

4

8

16

64

128

256

          k mk nhé

1+1=2

2+2=4

4+4=8

8+8=16

16+16=32

32+32=64

64+64=128

128+128=256

tk m nhé

X x (1/2+1/4+1/8+1/16+1/32+1/64+1/128) = 127/128

X x 127/128                                            = 127/128

                                                        X   = 127/128 : 127/128

                                                        X   = 1

1+1=2

2+2=4

4+4=8

8+8=16

16+16=32

32+32=64

64+64=128

128+128=256

2

4

8

16

32

64

128

256

1+1=2

2+2=44+4=8

8+8=16

16+16=32

32+32=64

64+64=128

128+128=256

\(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}=\frac{32+16+8+4+2+1}{128}=\frac{63}{128}\)

Đặt A=1/4+1/8+1/16+1/32+1/64+1/128.

\(A=\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^7}\)

\(2A=\left(\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^7}\right)\)

\(2A=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^6}\)

\(2A-A=\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^6}\right)-\left(\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^7}\right)\)

\(A=\frac{1}{2}-\frac{1}{2^7}\)

A=1/4+1/8+1/16+1/32+1/64+1/128

Ax2-A=(1/4+1/8+1/16+1/32+1/64+1/128)x2-(1/4+1/8+1/16+1/32+1/64+1/128)

Ax2-A=(1/2+1/4+1/8+1/16+1/32+1/64)-(1/4+1/8+1/16+1/32+1/64+1/128)

Ax2-A=1/2-1/4+1/4-1/8+1/8-1/16+1/16-1/32+1/32-1/64+1/64-1/128

A=1/2-1/128

A=    63/128.

b: A=1/3+1/9+...+1/3^10

=>3A=1+1/3+...+1/3^9

=>A*2=1-1/3^10=(3^10-1)/3^10

=>A=(3^10-1)/(2*3^10)

c: C=3/2+3/8+3/32+3/128+3/512

=>4C=6+3/2+...+3/128

=>3C=6-3/512

=>C=1023/512

d: A=1/2+...+1/256

=>2A=1+1/2+...+1/128

=>A=1-1/256=255/256

C= \(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\dfrac{1}{32}+\dfrac{1}{64}+\dfrac{1}{128}\)

2C = \(1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\dfrac{1}{32}+\dfrac{1}{64}\)

2C-C = \(1-\dfrac{1}{128}\)

C= \(\dfrac{127}{128}\)

C= 12+14+18+116+132+164+112821​+41​+81​+161​+321​+641​+1281​

2C = 1+12+14+18+116+132+1641+21​+41​+81​+161​+321​+641​

2C-C = 1−11281−1281​

C= 127128128127​
 

S=1/4+1/8+1/16+1/32+1/64+1/128

\(S=\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^7}\)

\(2S=2\left(\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^7}\right)\)

\(2S=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^6}\)

\(2S-S=\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^6}\right)-\left(\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^7}\right)\)

\(S=\frac{1}{2}-\frac{1}{2^7}\)

S=(1/2-1/4)+(1/4-1/8)+(1/8-1/16)+(1/16-1/32)+(1/32-1/64)+(1/64-1/128)

S=1/2-1/4+1/4-1/8+1/8-1/16+1/16-1/32+1/32-1/64+1/64-1/128

S=1/2-(1/4-1/4)+(1/8-1/8)+(1/16-1/16)+(1/32-1/32)+(1/64-1/64)-1/128

S=1/2-1/128

S=63/128

Đặt A=\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}\)

\(2A=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}\)

\(2A-A=1-\frac{1}{128}\)

\(A=\frac{127}{128}\)

\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}=\frac{127}{128}\)

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