If `alpha` is complex fifth root of unity and `(1+alpha +alpha^(2)+ alpha^(3))^(2005) = p + qalpha + ralpha^(2) + salpha^(3)` (where p,q,r,s are real), then find the value of `p+ q+r+s`.

Answer

`-1`

Explanation

`alpha ` is a complex fifth roots of unity.
`therefore 1+ alpha +alpha^(2) + alpha^(3)+ alpha^(4) = 0`
Now, given that
`(1+ alpha + alpha^(3) + alpha^(3))^(2005) = p + qalpha + ralpha^(2) + salpha^(3)`
`rArr (-alpha^(4)) ^(2005) = p + qalpha + ralpha^(2) + salpha^(3)`
`rArr - alpha^(8020) = p+ qalpha + ralpha^(2)+ salpha^(3)`
`rArr -(alpha^(5))^(1640) = p +qalpha+ = ralpha^(2) + salpha^(3)`
`rArr -1 = p + qalpha + ralpha^(2) + salpha^(3)`
`rArr p = - 1, q = r = s = 0`
`therefore p + q+ r+s = -1`