space z 1 comma space z 2 comma space z 3 comma space z 4 comma space z 5 comma space z 6 space a r e space t h e space r o o t s space o f space t h e space e q space space z to the power of 6 space plus z to the power of 5 plus z to the power of 4 plus z cubed plus space z squared plus z plus 1 equals 0 space. space t h e n space t h e space v a l u e space o f space sum from i equals 1 to 6 of z subscript i superscript 5 space plus sum from i equals 1 to 6 of z subscript i superscript 14 space minus stack product z subscript i space space space space space end subscript with i equals 1 below and 6 on topis

Asked by Sunil Soni | 26th May, 2016, 09:52: AM

Expert Answer:

begin mathsize 12px style z to the power of 6 plus z to the power of 5 plus z to the power of 4 plus z cubed plus z squared plus z plus 1 equals 0 rightwards double arrow fraction numerator z to the power of 7 minus 1 over denominator z minus 1 end fraction equals 0 rightwards double arrow z not equal to 1 space a n d space z to the power of 7 minus 1 equals 0 rightwards double arrow z to the power of 7 equals 1 rightwards double arrow z 1 comma space z 2 comma space z 3 comma space z 4 comma space z 5 space a n d space z 6 space a r e space c o m p l e x space 7 t h space r o o t s space o f space u n i t y. I f space z 1 equals alpha equals e to the power of i fraction numerator 2 straight pi over denominator 7 end fraction end exponent t h e space o t h e r space r o o t s space a r e space alpha squared comma space alpha cubed comma space.... alpha to the power of 6 sum from i equals 1 to 6 of z subscript i to the power of 5 equals alpha to the power of 5 plus alpha to the power of 10 plus alpha to the power of 15 plus alpha to the power of 20 plus alpha to the power of 25 plus alpha to the power of 30 equals open parentheses 1 plus alpha to the power of 5 plus alpha to the power of 10 plus alpha to the power of 15 plus alpha to the power of 20 plus alpha to the power of 25 plus alpha to the power of 30 close parentheses minus 1 equals fraction numerator open parentheses alpha to the power of 5 close parentheses to the power of 7 minus 1 over denominator alpha to the power of 5 minus 1 end fraction minus 1 equals fraction numerator open parentheses alpha to the power of 7 close parentheses to the power of 5 minus 1 over denominator alpha to the power of 5 minus 1 end fraction minus 1 equals 0 minus 1 equals negative 1 space left parenthesis sin c e space alpha to the power of 7 equals 1 right parenthesis sum from i equals 1 to 6 of z subscript i to the power of 14 equals alpha to the power of 14 plus alpha to the power of 28 plus alpha to the power of 14 cross times 3 end exponent plus alpha to the power of 14 cross times 4 end exponent plus alpha to the power of 14 cross times 5 end exponent plus alpha to the power of 14 cross times 6 end exponent equals open parentheses alpha to the power of 7 close parentheses squared plus open parentheses alpha to the power of 7 close parentheses to the power of 4 plus open parentheses alpha to the power of 7 close parentheses to the power of 6 plus open parentheses alpha to the power of 7 close parentheses to the power of 8 plus open parentheses alpha to the power of 7 close parentheses to the power of 10 plus open parentheses alpha to the power of 7 close parentheses to the power of 12 equals 1 plus 1 plus 1 plus 1 plus 1 plus 1 equals 6 stack product z subscript i equals with i minus 1 below and 6 on top alpha cross times alpha squared cross times alpha cubed cross times alpha to the power of 4 cross times alpha to the power of 5 cross times alpha to the power of 6 equals alpha to the power of 21 equals open parentheses alpha to the power of 7 close parentheses cubed equals 1 H e n c e comma space sum from i equals 1 to 6 of z subscript i to the power of 5 space plus space sum from i equals 1 to 6 of z subscript i to the power of 14 space minus stack product z subscript i with i minus 1 below and 6 on top equals negative 1 plus 6 plus 1 equals 6 end style

Answered by satyajit samal | 26th May, 2016, 02:01: PM

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