InterviewSolution
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InterviewSolution
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If `z_(r):r = 1,2,3,.....50` are the roots of the equaiton `sum_(r=0)^(50) z^(r)= 0`, then find the value of ` sum_(r=1)^(50)1//(z_(r) - 1)` |
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Answer» Correct Answer - `-25` `E = (1)/(z_(1)-1) + (1)/(z_(2)-1) +......+ (1)/(z_(50)-1)` Where `z_(1),z_(2),…..,z_(50)` are the roots of the equation `z^(51) -1=0` other than 1 Let . `1+z+z^(2)+……+z^(50)) = (z-z_(2)) …..(1-z_(50))` v`rArr log (1+z+z^(2) +…..+z^(50)) = log [(z-z_(1))(z -z_(2)).....(1-z_(50))]` Differentiating both sides w.r.t.s and putting z = 1 `(1+2z+3z^(2)+......+50_(z)^(49))/(1+z+z^(2) +......+z^(50))` `=(1)/(z-z_(1)) +(1)/(z-z_(2))+.......+(1)/(z-z_(50))` `rArr(50xx 51)/(2xx51)=-[(1)/(z_(1)-1)+(1)/(z_(2)-1)+......+(1)/(z_(50) -1)]` `therefore sum(1)/(z_(r)-1) = -25` |
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