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the least postitive interger n from ...

the least postitive interger n from which `root(3) (n+1) -root(3)(n)lt (1)/(12)` is -

A

6

B

7

C

8

D

9

Text Solution

Verified by Experts

The correct Answer is:
C
`(n+1)^(1//3)-n^(1//3) lt (1)/(12) `
` (n+1)-n-3(n+1)^(1//3)n^(1//3)-n^(1//3)lt ((1)/(12))^(3)`
` 1-3n^(1//3) (n+1)^(1//3)xx(1)/(12)lt (1)/((12)^(3))`
`(12)^(3) -3.(12)^(2) n^(1//3)(n+1)^(1//3) lt 1`
` (12)^(3) -1lt 3.(12)^(12)n^(1//3)(n+1)^(1//3)`
`(1727)/(3xx144)lt n^(1//3)(n+1)^(1//3)`
`n(n+1)gt((1727)/(3xx144))^(3)`
`n(n+1)gt 63.88`
`n=8`
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