The value of
` 6+ log_(3//2) (1/(3sqrt2)sqrt(4-1/(3sqrt2)sqrt(4-1/(3sqrt2)sqrt(4-1/(3sqrt2)...))))` is ________.

Options

A. 1
B. 2
C. 3
D. 4

Answer

D

Explanation

Let `sqrt(4-(1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2))….)))=y`. Then, `sqrt(4-(1)/(3sqrt(2))y) = y`
`rArr 4-(1)/(3sqrt(2)) y = y^(2)`
`rArr y^(2) + (1)/(3sqrt(2))y-4 = 0`
`rArr 3sqrt(2)y^(2) + y - 12sqrt(2) =0`
`rArr y = (-1+-sqrt(1+288))/(6sqrt(2)) = (-1 +-17)/(6sqrt(2))`
`rArr y = (16)/(6sqrt(2)) "or", y = (-18)/(6sqrt(2)) rArr y = (8)/(3sqrt(2)) " "[because y gt 0]`
`therefore 6+"log"_(3//2) {(1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2))….)))}`
` =6 +"log"_(3//2) ((1)/(3sqrt(2)) xx (8)/(3sqrt(2))) = 6 + "log"_(3//2) ((2)/(3))^(2) = 6-2=4`