If `omega` is an imaginary `x^n` root of unit then `underset(r=1)oversetnsum(ar+b)omega^(r-1)` is

Explanation

`E= sum_(r=1)^n(a+b) omega^(r-1)`

`= (a+b) +((2a+b) omega + (3a+b)omega^2 + (4a+b) omega^3 + .... + (na+b)omega^(n-1)`

`=b (1+ omega + omega^2 + .....+ omega^(n-1)) + a(1+ 2omega + 3omega^2 + 4 omega^3 + .... + n omega^(n-1)`

S`= 1 + 2omega + 3omega^2 + 4omega^3 + ..... + nomega^(n-2)`

S`omega= omega + 2omega^2 + 3omega^3 + .... + nomega^n`

subtracting these equations we get

`S(1-omega)= 1+ omega + omega^2 + omega^3 + .... + omega^(n-1) - n omega^n`

`= 0- n omega^n = -n`

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