Let `z_(1),z_(2),z_(3),……z_(n)` be the complex numbers such that `|z_(1)|= |z_(2)| = …..=|z_(n)| = 1`. ltbgt If `z = (sum_(k=1)^(n)Z_(k)) (sum_(k=1)^(n)(1)/(z_(k)))` then prove that (a) z is a real number (b) `0 lt z le n^(2)`

Explanation

`zbarz=1 or z =(1)/(z)`
Now`, z=(z_(1) +z_(2)+z_(3) + … + z_n ) ( (1)/(z _ 1 ) + (1)/(z_ 2 ) + … (1)/(z_n)) `
` = (z_ 1 + z _2 + … + z _n) (barz_1 + barz_2 + … + barz_n)`
` = ( z_1 + z_2 + … + z_n) (bar(z_1 + z_2 + ... + z _n)) `
` = | z _1 + z _2 + ... + z _n| ^(2) ` which is real (a)
` le (| z_1 | + |z_2| + ... + | z_n|) ^(2) = n ^(2)`
` therefore 0 lt z le n ^(2)`
Also, z is real number.