A=1+1/2+1/2^2+…..+1/2^100
B=1/3+1/3^2+1/3^3+…+1/3^80
C=1+1/4+1/4^2+…+1/4^120
D=-1/3+1/3^2-1/3^3+…-1/3^79+1/3^80
A=1+1/2+1/2^2+…..+1/2^100
B=1/3+1/3^2+1/3^3+…+1/3^80
C=1+1/4+1/4^2+…+1/4^120
D=-1/3+1/3^2-1/3^3+…-1/3^79+1/3^80
a) `A= 1 + 1/2 + 1/2^2 +…+1/2^100`
`2A = 2( 1+ 1/2+ 1/2^2+…+1/2^100)`
`2A= 2 + 1 + 1/2 +…+1/2^99`
`2A-A = 2+ 1+ 1/2+…+1/2^99 – 1- 1/2 – 1/2^2-…-1/2^100`
`A= 2 – 1/2^100`
Vậy `A= 2 – 1/2^100`
b) `B= 1/3 + 1/3^2 + 1/3^3+…+1/3^80`
`1/3 B= 1/3( 1/3 + 1/3^2 +1/3^3+….+1/3^80)`
`1/3 B= 1/3^2 + 1/3^3+ 1/3^4 +…+1/3^81`
`B- 1/3 B = 1/3 + 1/3^2 + 1/3^3+…+1/3^80 – 1/3^2 – 1/3^3 – 1/3^4 -…-1/3^81`
`2/3 B= 1/3 – 1/3^81`
`B= (1/3 – 1/3^81) : 2/3`
`B= (1/3 – 1/3^81). 3/2`
`B= 1/2 – 1/( 3^80 . 2)`
Vậy `B= 1/2 – 1/(3^80 .2)`
c) `C= 1 + 1/4 + 1/4^2+…+1/4^120`
`4C = 4(1+1/4 + 1/4^2 +…+1/4^120)`
`4C= 4+ 1 + 1/4 + ….+1/4^119`
`4C – C= 4 + 1 + 1/4 +…+1/4^119 – 1 – 1/4 – 1/4^2-…-1/4^120`
`3C= 4 – 1/4^120`
`C= ( 4- 1/4^120) :3`
`C= (4- 1/4)^120 . 1/3`
`C= 4/3 – 1/(4^120 .3)`
Vậy `C= 4/3 – 1/(4^120 .3)`
d) `D= -1/3 + 1/3^2 – 1/3^3 +…-1/3^79 + 1/3^80`
`1/3 D = 1/3( -1/3 + 1/3^2 – 1/3^3+…-1/3^79 + 1/3^80)`
`1/3 D= -1/3^2 + 1/3^3 – 1/3^4 +…- 1/3^80 + 1/3^81`
`D+ 1/3D = -1/3+ 1/3^2 – 1/3^3 +…-1/3^79 + 1/3^80 – 1/3^2 + 1/3^3 – 1/3^4 +…- 1/3^80 + 1/3^81`
`4/3 D= -1/3 + 1/3^81`
`D= (-1/3 + 1/3^81) : 4/3`
`D= (-1/3 + 1/3^81) . 3/4`
`D= -1/4 + 1/(3^80 .4)`
Vậy `D= -1/4 + 1/(3^80 .4)`
Đáp án:
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