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Complex Numbers 1+z+z 2 +z 3 +....z n-1 =(z-a 1 )(z-a 2 )(z-a n-1 ) HOW??

Complex Numbers
 
1+z+z2+z3+....zn-1=(z-a1)(z-a2)(z-an-1)
 
HOW??

Grade:12

1 Answers

Jayant Kishore
144 Points
8 years ago
Since 1, a1 , a2 , .......  are the roots of zn-1=0 therefore
zn-1 = (z-1)(z-a1)(z-a2)(z-a3).......(z - an-1)
=>  (zn-1)/(z-1) = (z-a1)(z-a2)(z-a3)....(z - an-1) 
=>  1+z+z2+z3+...... zn-1 =    (z-a1)(z-a2)(z-a3)....(z - an-1)    [ since (zn-1)/(z-1) is the sum of Geometric                                                                                                 Progression with 1st term 1 and common ratio z]
Hence Proved

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