Correct Answer - A
Let `x = sqrt(5sqrt(5sqrt(5 ... oo)))`
`x = sqrt(5x)`
`x^(2) = 5x`
`implies x = 5 (because x ne 0)`
`therefore log_(5) sqrt(5sqrt(5 ... oo)) = log_(5) 5 = 1`.
Consider `(1)/(2) + ((1)/(2))^(2) + ((1)/(2))^(3) + ... + oo`
Clearly it is a GP, `s_(oo) = (a)/(1-r)`
Here `a = (1)/(2)` and `r = (1)/(2)`
`=((1)/(2))/(1-(1)/(2))=1`
`therefore log ((1)/(2) + ((1)/(2))^(2) + ... ) = log 1 = 0`.
`therefore` The required value = 1 + 0 = 1.
