if |Z1| = |Z2| =.......=|Zn| = 1 prove that
 
|Z1 +  Z2 + Z3 +.....+Zn |  =  | 1/Z1 + 1/Z2 + 1/Z3 + ......+ 1/Zn |

Asked by ahuja8087 | 12th Sep, 2015, 12:37: PM

Expert Answer:

G i v e n space t h a t space open vertical bar z subscript 1 close vertical bar equals open vertical bar z subscript 2 close vertical bar equals open vertical bar z subscript 3 close vertical bar equals... equals open vertical bar z subscript n close vertical bar equals 1 W e space n e e d space t o space p r o v e space t h a t open vertical bar z subscript 1 plus z subscript 2 plus z subscript 3 plus... plus z subscript n close vertical bar equals open vertical bar 1 over z subscript 1 plus 1 over z subscript 2 plus 1 over z subscript 3 plus... plus 1 over z subscript n close vertical bar C o n s i d e r space open vertical bar z subscript 1 plus z subscript 2 plus z subscript 3 plus... plus z subscript n close vertical bar equals open vertical bar fraction numerator z subscript 1 top enclose z subscript 1 end enclose over denominator top enclose z subscript 1 end enclose end fraction plus fraction numerator z subscript 2 top enclose z subscript 2 end enclose over denominator top enclose z subscript 2 end enclose end fraction plus fraction numerator z subscript 3 top enclose z subscript 3 end enclose over denominator top enclose z subscript 3 end enclose end fraction plus... plus fraction numerator z subscript n top enclose z subscript n end enclose over denominator top enclose z subscript n end enclose end fraction close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open vertical bar fraction numerator open vertical bar z subscript 1 close vertical bar squared over denominator top enclose z subscript 1 end enclose end fraction plus fraction numerator open vertical bar z subscript 2 close vertical bar squared over denominator top enclose z subscript 2 end enclose end fraction plus fraction numerator open vertical bar z subscript 3 close vertical bar squared over denominator top enclose z subscript 3 end enclose end fraction plus... plus fraction numerator open vertical bar z subscript n close vertical bar squared over denominator top enclose z subscript n end enclose end fraction close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open vertical bar fraction numerator 1 over denominator top enclose z subscript 1 end enclose end fraction plus fraction numerator 1 over denominator top enclose z subscript 2 end enclose end fraction plus fraction numerator 1 over denominator top enclose z subscript 3 end enclose end fraction plus... plus fraction numerator 1 over denominator top enclose z subscript n end enclose end fraction close vertical bar space space space space space open square brackets because open vertical bar z subscript 1 close vertical bar equals open vertical bar z subscript 2 close vertical bar equals open vertical bar z subscript 3 close vertical bar equals open vertical bar z subscript 4 close vertical bar equals... equals open vertical bar z subscript n close vertical bar equals 1 close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open vertical bar top enclose 1 over z subscript 1 plus 1 over z subscript 2 plus 1 over z subscript 3 plus... plus 1 over z subscript n end enclose close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open vertical bar 1 over z subscript 1 plus 1 over z subscript 2 plus 1 over z subscript 3 plus... plus 1 over z subscript n close vertical bar

Answered by Vimala Ramamurthy | 12th Sep, 2015, 07:28: PM