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The sequence {x(k)} is defined by x(k+1)...

The sequence `{x_(k)}` is defined by `x_(k+1)=x_(k)^(2)+x_(k)` and `x_(1)=(1)/(2)`. Then `[(1)/(x_(1)+1)+(1)/(x_(2)+1)+...+(1)/(x_(100)+1)]` (where `[.]` denotes the greatest integer function) is equal to

A

`0`

B

`2`

C

`4`

D

`1`

Text Solution

Verified by Experts

The correct Answer is:
D
`(d)` `(1)/(x_(k+1))=(1)/(x_(k)(x_(k)+1))=(1)/(x_(k))-(1)/(x_(k)+1)`
`implies(1)/(x_(k)+1)=(1)/(x_(k))-(1)/(x_(k-1))`
`:. (1)/(x_(1)+1)+(1)/(x_(2)+1)+...+(1)/(x_(100)+1)=(1)/(x_(1))=(1)/(x_(101))`
As `0 lt (1)/(x_(101)) lt 1`
`:.[(1)/(x_(1)+1)+(1)/(x_(2)+1)+...+(1)/(x_(100)+1)]=1`
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