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The sum of first 20 terms of the sequenc...

The sum of first 20 terms of the sequence `0.7,0.77,0.777,"……"` is

A

`(7)/(9)(99-10^(-20))`

B

`(7)/(81)(179+10^(-20))`

C

`(7)/(9)(99+10^(-20))`

D

`(7)/(81)(179-10^(-20))`

Text Solution

Verified by Experts

The correct Answer is:
B
`S=0.7+0.77+0.777+"......."" upto 20 terms "`
`=(7)/(9)(0.9+0.99+0.999+"......."" upto 20 terms ")`
`=(7)/(9)[(1-0.1)+(1-0.01)+(1-0.001)+"......."" upto 20 terms "]`
`=(7)/(9)[(1+1+1+"......."" upto 20 terms ")-((1)/(10)+(1)/(10^(2))+(1)/(10^(3))+"......."" upto 20 terms ")]`
`=(7)/(9)[20-((1)/(10)(1-((1)/(10))^(20)))/(1-(1)/(10))]=(7)/(9)[(180-1+10^(-20))/(9)]`
`=(7)/(81)(179+10^(-20))`.
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