Let f be the fundamental frequency of the string . If the string is divided into three segments `l_(1) , l_(2)` and `l_(3)` such that the fundamental frequencies of each segments be `f_(1) , f_(2)` and `f_(3)` , respectively . Show that ` (1)/(f) = (1)/(f_(1)) + (1)/(f_(2)) + (1)/(f_(3))`

Explanation

For a fixed tension T and mass density `mu` ,frequency is inversely proportional to the string length i.e.
`f prop (1)/(l) implies f = (v)/(2l) implies l = (v)/(2f)`
For the first length segment `f_(1) = (v)/(2 l_(1)) implies l_(1) = (v)/(2f_(1))`
For the second length segment `f_(2) = (v)/(2l_(2)) implies l_(2) = (v)/(2f_(2))`
For the third length segment `f_(3) = (v)/(2l_(3)) implies l_(3) = (v)/(2f_(3))`
Therefore , the total length `l = l_(1) + l_(2) + l_(3)`
`(v)/(2f) = (v)/(2f_(1)) + (v)/(2f_(2)) + (v)/(2f_(3)) implies (1)/(f) = (1)/(f_(1)) + (1)/(f_(2)) + (1)/(f_(3))`