Calculate for z
Let z be the root of equation x 5 − 1 = 0 with z not equal to 1. compute the value of z 1 5 + z 1 6 + z 1 7 + … + z 5 0 .
The answer is 1.
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1 solution
How did u factorize the last step???
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My goal was to "extract" 1 + z + z 2 + z 3 + z 4 , knowing that this expression equals 0 . I then noticed I could create groups of five terms at a time by multiplying this expression by z 5 n , 3 ≤ n ≤ 9 , to account for all but the z 5 0 term in the original sequence. It's not the standard way to factorize, but it seemed to be the most efficient way to solve the problem.
Note first that z 5 − 1 = ( z − 1 ) ( z 4 + z 3 + z 2 + z + 1 ) , so if z 5 − 1 = 0 and z = 1 we have that
z 4 + z 3 + z 2 + z + 1 = 0 .
So we then have that z 1 5 + z 1 6 + z 1 7 + . . . . . . + z 5 0 =
( 1 + z + z 2 + z 3 + z 4 ) ( z 1 5 + z 2 0 + z 2 5 + z 3 0 + z 3 5 + z 4 0 + z 4 5 ) + z 5 0 = z 5 0 = ( z 5 ) 1 0 = 1 1 0 = 1 .